aa r X i v : . [ m a t h . R A ] F e b MAJORITY-CLOSED MINIONS OF BOOLEAN FUNCTIONS
ERKKO LEHTONEN
Abstract.
The 93 minions of Boolean functions stable under left compositionwith the clone of self-dual monotone functions are described. As an easyconsequence, all ( C , C )-stable classes of Boolean functions are determinedfor an arbitrary clone C and for any clone C containing the clone of self-dualmonotone functions. Introduction
We consider functions of several arguments from a set A to another set B , thatis, mappings of the form f : A n → B for some positive integer n . Given such afunction f , the functions that can be obtained from f by manipulation of arguments– permutation of arguments, introduction or deletion of fictitious arguments, andidentification of arguments – are called minors of f . Classes of functions closedunder formation of minors are called minor-closed classes or minions. Minions have been investigated from different points of view in the past decades.In universal algebra, minions arise naturally as sets of operations induced by theterms of height 1 on an algebra. As an analogue of the classical Galois connectionPol–Inv that characterizes clones, minor-closed classes were characterized by Pip-penger [7] as the closed classes of the Galois connection induced by the preservationrelation between functions and relation pairs. Minors and minions have recentlyemerged and played an important role in the analysis of the complexity of con-straint satisfaction problems (CSP), especially in a new variant known as promiseCSP (see the survey article by Barto, Bul´ın, Krokhin, and Oprˇsal [2]).Minions may satisfy additional closure conditions. A well-known example of thisidea are clones, that is, classes of operations that contain all projections and areclosed under composition. Aichinger and Mayr [1] defined clonoids as minor-closedclasses that are also stable under left composition with the operations of an algebra B = ( B, F ) (and hence under compositions with the clone of B ) and used themas a tool in showing that there is no infinite ascending chain of subvarieties of afinitely generated variety with an edge term. In full generality, we can considerclasses of functions that are stable under right compositions with a clone C on A and under left compositions with a clone C on B , in brief, ( C , C ) -stable classes.Couceiro and Foldes [4] characterized ( C , C )-stable classes with a specializationof Pippenger’s Galois connection between functions and relation pairs where thetwo relations are limited to invariants of the clones C and C .This inevitably leads us to the problem of describing the lattice of ( C , C )-stable classes – both the structure of the lattice and the classes themselves. Letus denote by L ( C ,C ) the lattice of ( C , C )-stable classes. It is known that thereare uncountably many minions, even when | A | = | B | = 2 (in which case thereare only countably many clones); hence an explicit description of all minions maybe unattainable. On the other hand, for some pairs of clones C and C , there Centro de Matem´atica e Aplicac¸˜oes, Faculdade de Ciˆencias e Tecnologia, Universi-dade Nova de Lisboa, Quinta da Torre, 2829-516 Caparica, Portugal
Date : February 4, 2021. JI ∗ L LSMS SM M ∗ M ∗ M Λ U ∞ MU MU U U VW ∞ MW MW W W Ω Ω ∗ Ω ∗ Ω Figure 1.
Post’s lattice.may be only a countable or finite number of ( C , C )-stable classes, and it may bepossible to describe all of them. A possible starting point for a systematical studyof lattices of ( C , C )-stable classes is suggested by a recent result due to Sparks [9]that provides us with the cardinality of the lattice of clonoids with a two-elementtarget algebra. It is noteworthy in this case that whether the closure system isfinite, countably infinite, or uncountable depends only on the clone of the targetalgebra and not on the set A (as long as | A | > n -ary operation f ∈ O B with n ≥ near-unanimityoperation if f ( x, . . . , x, y, x, . . . , x ) = x for all x, y ∈ B , where the single occurrenceof y can occur in any of the n argument positions. A ternary near-unanimityoperation is called a majority operation. A ternary operation f ∈ O B is called a Mal’cev operation if f ( y, y, x ) = f ( x, y, y ) = x for all x, y ∈ B . Theorem 1.1 (Sparks [9, Theorem 1.3]) . Let A be a finite set with | A | > , andlet B := { , } . Denote by J A the clone of projections on A , and let C be a cloneon B . Then the following statements hold. (i) L ( J A ,C ) is finite if and only if C contains a near-unanimity operation. (ii) L ( J A ,C ) is countably infinite if and only if C contains a Mal’cev operationbut no majority operation. (iii) L ( J A ,C ) has the cardinality of the continuum if and only if C contains nei-ther a near-unanimity operation nor a Mal’cev operation. As a first attempt of describing lattices of ( C , C )-stable classes, we considerclasses of Boolean functions (operations on { , } ); this is the simplest nontrivialcase, and Theorem 1.1 is applicable. In this paper we focus on clones containing the AJORITY-CLOSED MINIONS OF BOOLEAN FUNCTIONS 3 majority operation µ on { , } (the lattice of clones on { , } is shown in Figure 1).Recall that µ generates the clone SM of self-dual monotone functions, and denotethe clone of projections by J . Since µ is a near-unanimity operation, it followsfrom Sparks’s result that there are only a finite number of ( J , C )-stable classes ofBoolean functions for any clone C containing µ . Our goal is to refine this result andto explicitly describe the ( J , C )-stable classes for every clone C such that SM ⊆ C .Regarding statement (ii) of Theorem 1.1, a clone C on { , } contains a Mal’cevoperation but no majority operation if and only if L ⊆ C ⊆ L , where L and L denote the clone of idempotent linear functions and the clone of all linear functions,respectively. This situation was completely described in [5]; the case where C = h + i was settled earlier by Kreinecker [6, Theorem 3.12]. As for statement (iii), a clone C contains neither a near-unanimity operation nor a Mal’cev operation if and only if C is contained in one of the following clones: Λ = h∧ , , i , V = h∨ , , i , I ∗ = h¬ , i , W ∞ = h→i , U ∞ = h i .This paper is organized as follows. In Section 2, we provide the basic notions andpreliminary results that will be needed in the remaining sections. Furthermore, wedevelop some tools that are applicable to the study of ( C , C )-stable classes also ina more general setting. In Section 3, we define properties of Boolean functions thatare needed for presenting our results. Section 4 is dedicated to our main result andits proof: a complete description of the ( J , SM )-stable classes of Boolean functions.The proof has two parts. Firstly, we show that the given classes are ( J , SM )-stable;this is straightforward verification. The more difficult part of the proof is to showthat there are no further ( J , SM )-stable classes. From the description of the ( J , SM )-stable classes, we can determine rather easily the ( C , C )-stable classes for clones C and C , where C is arbitrary and SM ⊆ C ; this is done in Section 5. Weconclude the paper with some comments on topics for further research in Section 6.2. Preliminaries
General.
The set of nonnegative integers and the set of positive integers aredenoted by N and N + , respectively. For n ∈ N , let [ n ] := { i ∈ N | ≤ i ≤ n } .We denote tuples by bold letters and their components by the correspondingitalic letters, e.g., a = ( a , . . . , a n ). Since an n -tuple a is formally a mapping a : [ n ] → A , we may compose a with a map σ : [ m ] → [ n ], and the resulting map a ◦ σ : [ m ] → A is the m -tuple a ◦ σ = ( a σ (1) , . . . , a σ ( m ) ); we write simply a σ for a ◦ σ .We identify n -tuples over A with words of length n over A . For a ∈ A and n ∈ N , a n stands for the word consisting of n copies of a .2.2. Functions of several arguments, function class composition, minors,and stability.
Let A and B be sets. We consider functions of several arguments from A to B , that is, mappings f : A n → B for some number n ∈ N + called the arity of f . Functions of several arguments from A to A are called operations on A .We denote the set of all functions of several arguments from A to B by F AB andthe set of all operations on A by O A . For any C ⊆ F AB and n ∈ N , we denote by C ( n ) the set of all n -ary members of C . In particular, F ( n ) AB and O ( n ) A are the setsof all n -ary functions of several arguments from A to B and the set of all n -aryoperations on A , respectively.For n ∈ N + and i ∈ [ n ], the i -th n -ary projection on A is the operationpr ( n ) i : A n → A , ( a , . . . , a n ) a i . The first unary projection pr (1)1 is thus theidentity map on A and is also denoted by id A or simply by id if the set A is clearfrom the context. We denote the set of all projections on A by J A . MAJORITY-CLOSED MINIONS OF BOOLEAN FUNCTIONS
Let A , B , and C be nonempty sets, and let f ∈ F ( n ) BC and g , . . . , g n ∈ F ( m ) AB .The composition of f with g , . . . , g n is the function f ( g , . . . , g n ) ∈ F ( m ) AC definedby the rule f ( g , . . . , g n )( a ) := f ( g ( a ) , . . . , g n ( a )) for all a ∈ A m . The notion ofcomposition extends to function classes as follows. Let I ⊆ F BC and J ⊆ F AB .The composition of I with J is the class IJ ⊆ F AC given by IJ := { f ( g , . . . , g n ) | n, m ∈ N , f ∈ I ( n ) , g , . . . , g n ∈ J ( m ) } . Let f ∈ F ( n ) AB , and let J A be the clone of projections on A . The functions in { f } J A are called minors of f . Thus a function g ∈ F ( m ) AB is a minor of f if andonly if g = f (pr ( m ) σ (1) , . . . , pr ( m ) σ ( n ) ) for some σ : [ n ] → [ m ]. We use the shorthand f σ for f (pr ( m ) σ (1) , . . . , pr ( m ) σ ( n ) ). Thus f σ ( a ) = f ( a σ ). For a map σ : [ n ] → [ m ], we definethe map σ : A m → A n by σ ( a ) := a σ . Then we can also write f σ = f ◦ σ .As a further notational tool, we may specify a map σ : [ n ] → [ m ] by the word σ (1) σ (2) . . . σ ( n ); thus we may write f σ (1) σ (2) ...σ ( n ) for f σ . The arity of the minoris not explicit in this notation but will be clear from the context.A set C ⊆ O A is called a clone on A if J A ⊆ C and CC ⊆ C . The set of allclones on A constitutes a closure system. The smallest and the greatest clones on A are the clone J A of projections and the clone O A of all operations. We denoteby h F i the clone generated by F , i.e., the smallest clone on A that contains F .Let F ⊆ F AB , and let C and C be clones on A and B , respectively. We saythat F is stable under right composition with C if F C ⊆ F , and we say that F is stable under left composition with C if C F ⊆ F . We say that F is ( C , C ) -stable if it is stable under right composition with C and stable under left compositionwith C . The set of all ( C , C )-stable classes constitutes a closure system. Wedenote by h F i ( C ,C ) the ( C , C )-stable class generated by F , i.e., the smallest( C , C )-stable class containing F . Lemma 2.1.
Let C and C ′ be clones on A and let C and C ′ by clones on B . If C ⊆ C ′ and C ⊆ C ′ , then every ( C ′ , C ′ ) -stable class is ( C , C ) -stable.Proof. Let F ⊆ F AB , and assume that F is ( C ′ , C ′ )-stable. We have F C ⊆ F C ′ ⊆ F and C F ⊆ C ′ F ⊆ F , so F is ( C , C )-stable. (cid:3) Lemma 2.2.
Let C and K be clones on A . Then the following statements hold: (i) KC ⊆ K if and only if C ⊆ K . (ii) CK ⊆ K if and only if C ⊆ K .Proof. If C ⊆ K , then KC ⊆ KK ⊆ K and CK ⊆ KK ⊆ K . Assume nowthat C * K . Then there exists an f ∈ C such that f / ∈ K . Since id ∈ K , wehave f = id( f ) ∈ KC and f = f (id , . . . , id n ) ∈ CK . Therefore KC * K and CK * K . (cid:3) Lemma 2.3 (Couceiro, Foldes [3, 4, Associativity Lemma]) . Let A , B , C , and D be arbitrary nonempty sets, and let I ⊆ F CD , J ⊆ F BC , and K ⊆ F AB . Thefollowing statements hold. (i) ( IJ ) K ⊆ I ( JK ) . (ii) If J is stable under right composition with the clone of projections on B (i.e., minor-closed), then ( IJ ) K = I ( JK ) . Lemma 2.4.
Let F ⊆ F AB , and let C and C be clones on A and B , respectively.Then h F i ( C ,C ) = C ( F C ) .Proof. Since the clones C and C contain all projections, we have that F ⊆ F C ⊆ C ( F C ). Applying the Associativity Lemma (Lemma 2.3), we see that C ( F C ) is AJORITY-CLOSED MINIONS OF BOOLEAN FUNCTIONS 5 ( C , C )-stable (note that clones are minor-closed and closed under composition):( C ( F C )) C ⊆ C (( F C ) C ) ⊆ C ( F ( C C )) ⊆ C ( F C ) ,C ( C ( F C )) = ( C C )( F C ) ⊆ C ( F C ) . Therefore C ( F C ) is a ( C , C )-stable class containing F ; hence the inclusion h F i ( C ,C ) ⊆ C ( F C ) holds. The converse inclusion follows immediately from the( C , C )-stability of h F i ( C ,C ) : C ( F C ) ⊆ C ( h F i ( C ,C ) C ) ⊆ C h F i ( C ,C ) ⊆ h F i ( C ,C ) . (cid:3) The following two helpful lemmata were proved in [5]. Here we employ thebinary composition operation ∗ defined as follows: if f ∈ O ( n ) A and g ∈ O ( m ) A , then f ∗ g ∈ O ( m + n − A is defined by( f ∗ g )( a , . . . , a m + n − ) := f ( g ( a , . . . , a m ) , a m +1 , . . . , a m + n − ) , for all a , . . . , a m + n − ∈ A . Lemma 2.5.
Let F ⊆ O A . Let C be a clone on A , and let G be a generating setof C . Then the following conditions are equivalent. (i) F C ⊆ F (ii) F is minor-closed and f ∗ g ∈ F whenever f ∈ F and g ∈ C . (iii) F is minor-closed and f ∗ g ∈ F whenever f ∈ F and g ∈ G . Lemma 2.6.
Let F ⊆ O A . Let C be a clone on A , and let G be a generating setof C . Then the following conditions are equivalent. (i) CF ⊆ F (ii) g ( f , . . . , f n ) ∈ F whenever g ∈ C ( n ) and f , . . . , f n ∈ F ( m ) for some n, m ∈ N . (iii) g ( f , . . . , f n ) ∈ F whenever g ∈ G ( n ) and f , . . . , f n ∈ F ( m ) for some n, m ∈ N . Terms and algebras, stratified terms.
Let X = { x , x , . . . } be a count-ably infinite set of variables, and for each n ∈ N , let X n := { x , . . . , x n } . Let F be a set of function symbols, and assume that F is disjoint from X . Assign toeach f ∈ F a natural number ar( f ), called the arity of f . The map τ : F → N , f ar( f ), is called an ( algebraic similarity ) type (also known as a signature ). Terms of type τ over X are defined as usual: every variable x i ∈ X is a term oftype τ , and if f ∈ F is an n -ary function symbol and t , . . . , t n are terms of type τ , then f ( t , . . . , t n ) is a term of type τ . Terms are thus certain well-formed wordsover the alphabet comprising the variables X , the function symbols F , and somepunctuation (parentheses and comma). Any subword of a term t that is itself aterm is called a subterm of t . If a term t is of the form f ( t , . . . , t n ), we call it functional; f is called the leading function symbol of t , and the terms t , . . . , t n arecalled the immediate subterms of t . We call terms of type τ over X also F -terms .We denote the set of all F -terms by T ( F ).The height of a term t , denoted by h ( t ), is defined recursively as follows: variableshave height 0 ( h ( x i ) := 0 for every x i ∈ X ), and if t = f ( t , . . . , t n ), then h ( t ) :=max( h ( t ) , . . . , h ( t n )) + 1.The set of variables occurring in a term t is denoted by var( t ). If t is a termwith var( t ) ⊆ X n and t , . . . , t n are terms, then we define t [ t , . . . , t n ] as the termobtained by substitution of t i for x i in t , i.e., by simultaneously replacing eachoccurrence of x i in t by the term t i , for each i ∈ [ n ].Let G and F be sets of function symbols, not necessarily disjoint. A term t ∈ T ( G ∪ F ) is ( G, F ) -stratified if the following conditions hold: MAJORITY-CLOSED MINIONS OF BOOLEAN FUNCTIONS (i) for every subterm t ′ of t with h ( t ′ ) = 1, the leading functional symbol of t ′ belongs to F ;(ii) for every subterm t ′ of t with h ( t ′ ) ≥
2, the leading functional symbol of t ′ belongs to G and no immediate subterm of t ′ is a variable.In other words, ( G, F )-stratified terms are exactly the terms of the form t [ t , . . . , t n ]for some t ∈ T ( G ) with var( t ) ⊆ X n , n ∈ N + , and t , . . . , t n ∈ T ( F ) such that h ( t i ) = 1 for all i ∈ [ n ]. We denote by Str( G, F ) the set of all (
G, F )-stratifiedterms in T ( G ∪ F ).Given a type τ : F → N , an algebra of type τ is a pair A = ( A, F A ), where A is a nonempty set called the universe and F A = ( f A ) f ∈ F is an indexed family ofoperations on A , where for each f ∈ F , the operation f A has arity ar( f ). For an n -ary term t of type τ and an algebra A = ( A, F ) of type τ , the term operation induced by t on A , denoted by t A , is the n -ary operation on A that is definedrecursively as follows. If t = x i ∈ X n , then t A := pr ( n ) i . If t = f ( t , . . . , t n ), where f is an n -ary operation symbol and t , . . . , t n are terms, then t A := f A ( t A , . . . , t A n ).We extend the notation to sets of terms: if T ⊆ T ( F ), then T A := { t A | t ∈ T } .Henceforth, we will regard functions as function symbols, and we give standardinterpretation to terms of the corresponding type. More precisely, we will assumethat F is a set of operations on A , and we consider the algebraic similarity type τ : F → N , f ar( f ), and the algebra A = ( A, F A ), where f A = f for each f ∈ F .We will then consider term operations induced by F -terms on A .It is well known that the set of term operations of an algebra is a clone, i.e., if F ⊆ O A and A = ( A, F ), then h F i = T ( F ) A . In fact, every clone arises in thisway. We now present an analogous description of the term operations induced bystratified terms, which will be called stratified term operations. Lemma 2.7.
Let G and F be sets of operations on A , let C = h G i , and let J A bethe clone of projections on A . Then h F i ( J A ,C ) = Str( G, F ) A .Proof. Assume first that f ∈ h F i ( J A ,C ) . By Lemma 2.4, f ∈ C ( F J A ), so there arefunctions g ∈ C and h , . . . , h n ∈ F J A such that f = g ( h , . . . , h n ). For each i ∈ [ n ],the function h i is a minor of some function from F , so there exists a terms t i ∈ T ( F )of height 1 representing h i . Since g is a member of the clone generated by G , thereexists a term t ∈ T ( G ) representing g . Then t [ t , . . . , t n ] is a ( G, F )-stratified termrepresenting g ( h , . . . , h n ) = f , so f ∈ Str(
G, F ) A .Conversely, assume that f ∈ Str(
G, F ) A . Then there exists a ( G, F )-stratifiedterm t [ t , . . . , t n ] representing f ; here t ∈ T ( G ) and t , . . . , t n ∈ T ( F ) are termsof height 1. Then t A ∈ h G i = C and t A , . . . , t A n ∈ F J A , so f = t A ( t A , . . . , t A n ) ∈ C ( F J A ) = h F i ( J A ,C ) by Lemma 2.4. (cid:3) Stratified term operations may fail to constitute a clone. The following twolemmata describe useful situations where this nevertheless happens.
Lemma 2.8.
Let G and F be sets of operations on A , let C = h G i and C ′ = h G ∪ F i . Then h F i ( J A ,C ) = C ′ if and only if for every term t ∈ T ( G ∪ F ) thereexists a ( G, F ) -stratified term t ′ ∈ Str(
G, F ) such that t A = ( t ′ ) A .Proof. Assume first that h F i ( J A ,C ) = C ′ , We have C ′ = T ( G ∪ F ) A and h F i ( J A ,C ) =Str( G, F ) A by Lemma 2.7, so clearly for every t ∈ T ( G ∪ F ) there exists t ′ ∈ Str(
G, F ) such that t A = ( t ′ ) A .Assume now that for every t ∈ T ( G ∪ F ) there exists a t ′ ∈ Str(
G, F ) suchthat t A = ( t ′ ) A . Then T ( G ∪ F ) A ⊆ Str(
G, F ) A . Since Str( G, F ) ⊆ T ( G ∪ F ),the converse inclusion Str( G, F ) A ⊆ T ( G ∪ F ) A clearly holds. Then h F i ( J A ,C ) =Str( G, F ) A = T ( G ∪ F ) A = C ′ . (cid:3) AJORITY-CLOSED MINIONS OF BOOLEAN FUNCTIONS 7
Lemma 2.9.
Let G and F be sets of operations on A . If (i) id A ∈ F , and (ii) for every ϕ ∈ F ( n -ary ) , α ∈ G ∪ F ( m -ary ) , and i ∈ [ n ] , there exists aterm t ∈ Str(
G, F ) equivalent to ϕ ( x , . . . , x i − , α ( x i , . . . , x i + m − ) , x i + m , . . . , x m + n − ) such that no variable is repeated in any subterm of t of height ,then for every term t ∈ T ( G ∪ F ) there exists a term t ′ ∈ Str(
G, F ) such that t A = ( t ′ ) A ; consequently h F i ( J A ,C ) = h G ∪ F i .Proof. We prove the claim by induction on the number of function symbols in aterm t ∈ T ( G ∪ F ). If t has no function symbol, then t = x i ∈ X , and we canchoose t ′ := id A ( x i ). If t has one function symbol, then t = f ( x i , . . . , x i n ) for some f ∈ G ∪ F . If f ∈ F , then t ∈ Str(
G, F ) and we are done. If f ∈ G , then t isequivalent to the term f (id A ( x i ) , . . . , id A ( x i n )), which is ( G, F )-stratified.Assume that the claim is true for every term with at most k function symbols( k ≥ t has k + 1 function symbols. Then t = f ( t , . . . , t n ) for some f ∈ G ∪ F . By the induction hypothesis, for every i ∈ [ n ],there exists a term t ′ i ∈ Str(
G, F ) such that t A i = ( t ′ i ) A . If f ∈ G , then t ′ := f ( t ′ , . . . , t ′ n ) ∈ Str(
G, F ) and t A = ( t ′ ) A .Assume now that f ∈ F . Since t contains at least two function symbols, at leastone of the immediate subterms of t is functional, say t i = α ( u , . . . , u m ) for some α ∈ G ∪ F . Let s := f ( x , . . . , x i − , α ( x i , . . . , x i + m − ) , x i + m , . . . , x m + n − ) . By our assumptions, there exists a term s ′ ∈ Str(
G, F ) such that s A = ( s ′ ) A and no variable is repeated in any subterm of s ′ of height 1. We clearly have t = s [ t , . . . , t i − , u , . . . , u m , t i +1 , . . . , t n ], and therefore t is equivalent to the term r := s ′ [ t , . . . , t i − , u , . . . , u m , t i +1 , . . . , t n ]. The term r is not necessarily ( G, F )-stratified, but we obtain an equivalent (
G, F )-stratified term as follows. Let q be asubterm of s ′ of height 1, so q = ϕ ( x i , . . . , x i k ) for some ϕ ∈ F and the variables x i , . . . , x i k are distinct. Then q [ t ′ i , . . . , t ′ i k ], where t ′ j := t j , if 1 ≤ j ≤ i − u j − i +1 , if i ≤ j ≤ i + m − t j − m +1 , if i + m ≤ j ≤ m + n − r and it has fewer function symbols than f , so by the inductionhypothesis there is an equivalent term q ′ ∈ Str(
G, F ). By replacing q [ t ′ i , . . . , t ′ i k ]by q ′ , for each subterm q of s ′ of height 1, we obtain the desired ( G, F )-stratifiedterm equivalent to t .It now follows from Lemma 2.8 that h F i ( J A ,C ) = h G ∪ F i . (cid:3) Remark 2.10.
When verifying the conditions of Lemma 2.9, it is not necessaryto consider the cases when ϕ = id A or α = id A , because the equivalent term t ′ ∈ Str(
G, F ) always exists in these cases, as specified below:id A ( α ( x , . . . , x m )) ≡ α (id A ( x ) , . . . , id A ( x m )) ,ϕ ( x , . . . , x i − , id A ( x i ) , x i +1 , . . . , x n ) ≡ ϕ (id A ( x ) , . . . , id A ( x n )) , id A (id A ( x )) ≡ id A ( x ) . Moreover, we may be able to further reduce the number of cases to consider bymaking use of possible symmetries of ϕ . More precisely, assume that the identity MAJORITY-CLOSED MINIONS OF BOOLEAN FUNCTIONS x ¬ x x ∧ ∨ + ↔ x x x µ ⊕ Figure 2.
Some well-known Boolean functions. ϕ ( x , . . . , x n ) ≈ ϕ ( x σ (1) , . . . , x σ ( n ) ) is satisfied by A for some permutation σ of [ n ].If ϕ ( x , . . . , x i − , α ( x i , . . . , x i + m − ) , x i + m , . . . , x m + n − )is equivalent to t ∈ Str(
G, F ), then, for j := σ − ( i ), the term ϕ ( x , . . . , x j − , α ( x j , . . . , x j + m − ) , x j + m , . . . , x m + n − )is equivalent to ϕ ( x ζ (1) , . . . , x ζ ( i − , α ( x ζ ( i ) , . . . , x ζ ( i + m − ) , x ζ ( i + m ) , . . . , x ζ ( m + n − ) , where ζ is the permutation of [ m + n −
1] given by the rule ζ ( k ) = σ ( k ) if k ≤ j − σ ( k ) ≤ i − σ ( k ) + m − k ≤ j − σ ( k ) ≥ i + 1, k + i − j if j ≤ k ≤ j + m − σ ( k − m + 1) if k ≥ j + m and σ ( k ) ≤ i − σ ( k − m + 1) + m − k ≥ j + m and σ ( k ) ≥ i + 1,and this term is equivalent to t [ x ζ (1) , x ζ (2) , . . . , x ζ ( m + n − ] ∈ Str(
G, F ).3.
Properties and classes of Boolean functions
Operations on { , } are called Boolean functions.
We will often encounter thefollowing well-known Boolean functions, defined by the operation tables in Figure 2:the constant functions 0 and 1, identity id, negation ¬ , conjunction ∧ , disjunction ∨ , addition +, equivalence ↔ , nonimplication , majority µ , and triple sum ⊕ .For a ∈ { , } n , let a := 1 − a . For a = ( a , . . . , a n ) ∈ { , } n , let a :=( a , . . . , a n ). We regard the set { , } totally ordered by the natural order 0 < { , } n . The poset ( { , } n , ≤ ) con-stitutes a Boolean lattice, i.e., a complemented distributive lattice with least andgreatest elements = (0 , . . .
0) and = (1 , . . . , a a beingthe complementation.Denote by Ω the class of all Boolean functions. For any C ⊆ Ω and any a, b ∈{ , } , let C a ∗ := { f ∈ C | f ( ) = a } , C ∗ b := { f ∈ C | f ( ) = b } ,C ab := C a ∗ ∩ C ∗ b ,C = := { f ∈ C | f ( ) = f ( ) } , C = := { f ∈ C | f ( ) = f ( ) } ,C ≤ := { f ∈ C | f ( ) ≤ f ( ) } , C ≥ := { f ∈ C | f ( ) ≥ f ( ) } ,C = , := C = ∪ C , C = , := C = ∪ C . Let f : { , } n → { , } . The negation f , the inner negation f n , and the dual f d of f are the n -ary Boolean functions given by the rules f ( a ) := f ( a ), f n ( a ) := f ( a ), AJORITY-CLOSED MINIONS OF BOOLEAN FUNCTIONS 9 and f d ( a ) := f ( a ), for all a ∈ { , } n . A function f : { , } n → { , } is reflexive if f = f n , i.e., f ( a ) = f ( a ) for all a ∈ { , } n , and f is self-dual if f = f d , i.e., f ( a ) = f ( a ) for all a ∈ { , } n . We denote by R and S the classes of all reflexiveand all self-dual functions, respectively. For any C ⊆ Ω , let C := { f | f ∈ C } , C n := { f n | f ∈ C } , and C d := { f d | f ∈ C } .Let f, g : { , } n → { , } . We say that f is a minorant of g if f ( a ) ≤ g ( a ) for all a ∈ { , } n , and we say that f is a majorant of g if f ( a ) ≥ g ( a ) for all a ∈ { , } n .We denote by S − and S + the classes of all minorants of self-dual functions andall majorants of self-dual functions, respectively. Note that S = S − ∩ S + and that f ∈ S − if and only if f ( a ) ∧ f ( a ) = 0 for all a ∈ { , } n , and f ∈ S + if and only if f ( a ) ∨ f ( a ) = 1 for all a ∈ { , } n .A function f ∈ { , } n → { , } is monotone if f ( a ) ≤ f ( b ) whenever a ≤ b .We denote by M the class of all monotone functions.A function f ∈ { , } n → { , } is 1 -separating of rank k if for all a , . . . , a k ∈ f − (1) it holds that a ∧ · · · ∧ a k = , and f is 0 -separating of rank k if for all a , . . . , a k ∈ f − (0) it holds that a ∨ · · · ∨ a k = . We denote by U k and W k theclasses of all 1-separating functions of rank k and all 0-separating functions of rank k . We introduce the shorthands SM for S ∩ M , MU k for M ∩ U k , and MW k for M ∩ W k .We denote by C the class of all constant functions, and we introduce the short-hands C := C and C := C . We denote by J the class of all projections on { , } .The clones of Boolean functions were described by Post [8]. In this paper, wewill need only a handful of them, namely the clones Ω , Ω ∗ , Ω ∗ , Ω , M , M ∗ , M ∗ , M , S , S , SM , U , U , MU , MU , W , W , MW , MW , and J thatwere defined above. The lattice of clones of Boolean functions, Post’s lattice, ispresented in Figure 1, and the above-mentioned clones are indicated in the Hassediagram. 4. ( J , SM ) -stable classes Theorem 4.1.
There are precisely 93 ( J , SM ) -stable classes of Boolean functions,and they are the following: Ω , Ω ≤ , Ω ≥ , Ω = , , Ω = , , Ω ∗ ∪ C , Ω ∗ ∪ C , Ω ∪ C , Ω ∪ C , Ω ∪ C , Ω ∪ C , Ω ∗ ∪ C , Ω ∗ ∪ C , Ω ∪ C , Ω ∪ C , Ω ∪ C , Ω ∪ C , Ω = , Ω = , Ω ∗ , Ω ∗ , Ω ∗ , Ω ∗ , Ω , Ω , Ω , Ω , M , M ∗ , M ∗ , M , M , M ∗ , M ∗ , M , S − , S −6 = , S − ∗ , S −∗ , S − ∪ C , S − ∪ C , S − , S − , S − , S + , S + = , S +1 ∗ , S + ∗ , S +10 ∪ C , S +01 ∪ C , S +10 , S +01 , S +11 , S , S , S , SM , SM , U , U ∪ C , U , MU , MU , U , U , U ∪ C , U , MU , MU , U , W , W ∪ C , W , MW , MW , W , W , W ∪ C , W , MW , MW , W , U ∩ W , W ∩ U , R , R ∪ C , R ∪ C , R , R , C , C , C , ∅ . The lattice of ( J , SM )-stable classes is shown in Figure 3. The Hasse diagram isdrawn in such a way that the reflection along the central vertical axis correspondsto the automorphism C C .The remainder of this section is devoted to the proof of Theorem 4.1. The proofcomprises two main parts. Firstly, we show that each class listed in the theorem is( J , SM )-stable; this is rather straightforward verification. Secondly, we prove that ∅ C C CU ∩ W W ∩ U U U W W SM SMM M ∗ M ∗ MMU M U U U ∪ C U MW MW W W ∪ C W M M ∗ M ∗ M MU M U U U ∪ C U MW MW W W ∪ C W ΩΩ ≤ Ω = , Ω = , Ω ≥ Ω = Ω = Ω ∗ ∪ C Ω ∗ Ω ∗ ∪ CΩ ∗ Ω ∗ ∪ CΩ ∗ Ω ∗ ∪ C Ω ∗ Ω Ω ∪ C Ω ∪ C Ω ∪ C Ω Ω ∪ C Ω ∪ C Ω ∪ C SS S S − S + S −6 = S + = S − ∗ S +1 ∗ S + ∗ S −∗ S − S − ∪ C S + ∪ C S +10 S +01 S +01 ∪ C S − ∪ C S − S − S +11 Ω Ω ∪ C Ω Ω ∪ CR R ∪ C R ∪ C R R Figure 3. ( J , SM )-stable classes. Clones are highlighted in red.there are no further ( J , SM )-stable classes. This is the more complicated part ofthe proof. The idea is to show that any set of Boolean functions generates oneof the ( J , SM )-stable classes listed in Theorem 4.1, namely the one suggested byFigure 3. We develop some tools that allow us to easily determine what a given setof functions generates.Without further ado, we now show that the classes listed in Theorem 4.1 are( J , SM )-stable. With the help of the following two lemmata, we can reduce thenumber of classes we need to consider. Lemma 4.2.
Let
C, F ⊆ Ω , and assume that h F i ( J , SM ) = C . Then also C , C n ,and C d are ( J , SM ) -stable and h F i ( J , SM ) = C , h F n i ( J , SM ) = C n , h F d i ( J , SM ) = C d .Proof. In order to show that C is ( J , SM )-stable, take three arbitrary members of C ; they are of the form f , g, h for some f, g, h ∈ C . Then f σ = (id ◦ f ) ◦ σ =id ◦ ( f ◦ σ ) = f σ ∈ C because f σ ∈ C . Moreover, µ ( f , g, h ) = µ ( f, g, h ) ∈ C because µ ( f, g, h ) ∈ C . We conclude that C is ( J , SM )-stable. AJORITY-CLOSED MINIONS OF BOOLEAN FUNCTIONS 11
In order to show that C is generated by F , let f ∈ C be n -ary. Then f ∈ C , sothere is a function g ∈ SM ( m ) , h , . . . , h m ∈ F and n -ary minors ( h i ) σ i (1 ≤ i ≤ m )such that f = g (( h ) σ , . . . , ( h m ) σ m ). But then f = g (( h ) σ , . . . , ( h m ) σ m ) = g (( h ) σ , . . . , ( h m ) σ m )= g (( h ) σ , . . . , ( h m ) σ m ) = g (( h ) σ , . . . , ( h m ) σ m ) ∈ h F i ( J , SM ) , where the third equality holds because g is self-dual. Therefore C ⊆ h F i ( J , SM ) .Since F ⊆ C and C is ( J , SM )-stable, we have h F i ( J , SM ) ⊆ h C i ( J , SM ) = C .In order to show that C n is ( J , SM )-stable, take three arbitrary members of C n ;they are of the form f n , g n , h n for some f, g, h ∈ C . Then ( f n ) σ ( a , . . . , a m ) = f ( a σ (1) , . . . , a σ ( n ) ) = ( f σ ) n ( a , . . . , a m ), i.e., ( f n ) σ = ( f σ ) n ∈ C n because f σ ∈ C .Moreover, µ ( f n , g n , h n ) = ( µ ( f, g, h )) n ∈ C n because µ ( f, g, h ) ∈ C . We concludethat C n is ( J , SM )-stable.In order to show that C n is generated by F n , let f n ∈ C n be n -ary. Then f ∈ C ,so there is a function g ∈ SM , h , . . . , h m ∈ F , and n -ary minors ( h i ) σ i (1 ≤ i ≤ m )such that f = g (( h ) σ , . . . , ( h m ) σ m ). But then f n = ( g (( h ) σ , . . . , ( h m ) σ m )) n = g ((( h ) σ ) n , . . . , (( h m ) σ m ) n )= g (( h n1 ) σ , . . . , ( h n m ) σ m ) ∈ h F n i ( J , SM ) , so C n ⊆ h F n i ( J , SM ) . Since F n ⊆ C n and C n is ( J , SM )-stable, we have h F n i ( J , SM ) ⊆h C n i ( J , SM ) = C n .The statements about C d and F d follow from the above, because f d = f n = ( f ) n for any f ∈ Ω . (cid:3) Lemma 4.3.
Let C be a ( J , SM ) -stable class. (i) If {∧} C ⊆ C then C ∪ C is ( J , SM ) -stable. (ii) If {∨} C ⊆ C then C ∪ C is ( J , SM ) -stable. (iii) If {∧ , ∨} C ⊆ C then C ∪ C is ( J , SM ) -stable.Proof. (i) Assume that C is ( J , SM )-stable and {∧} C ⊆ C . Then C ∪ C is clearlyminor-closed. Since SM = h µ i , by Lemma 2.6, it remains to show that C ∪ C isstable under left composition with { µ } . Let f, g, h ∈ C ∪ C , all n -ary. If f , g , and h are all in C , then µ ( f, g, h ) ∈ C because C is stable under left composition with SM . If two of f , g , and h are in C and the third is in C (that is, it is a constant0 function), say f, g ∈ C and h = 0, then µ ( f, g, h ) = µ ( f, g,
0) = f ∧ g ∈ C . If(at least) two of f , g , and h are in C , say g = 0 and h = 0, then µ ( f, g, h ) = µ ( f, ,
0) = 0 ∈ C . We conclude that C ∪ C is ( J , SM )-stable.(ii) The proof is similar to that of (i) and makes use of the fact that µ ( f, g,
1) = f ∨ g .(iii) The proof is similar to the previous parts. The only new case to consideris when f, g, h ∈ C ∪ C and there are two different constant functions among f , g ,and h , say g = 0 and h = 1. Then µ ( f, g, h ) = µ ( f, ,
1) = f ∈ C ∪ C . We concludethat C ∪ C is ( J , SM )-stable. (cid:3) Recall that an operation f ∈ O ( n ) A preserves a relation ρ ⊆ A m if for all( a i , . . . , a im ) ∈ ρ ( i ∈ [ n ]), we have ( f ( a , . . . , a n ) , . . . , f ( a m , . . . , a nm )) ∈ ρ . Lemma 4.4.
The fuction µ preserves every binary relation on { , } .Proof. Every function preserves the empty relation, so assume ∅ 6 = ρ ⊆ { , } ,and let ( a , b ) , ( a , b ) , ( a , b ) ∈ S . Then there exist p, q, r, s ∈ [3] such that p = q , r = s , a p = a q , and b r = b s . We have µ ( a , a , a ) = a p = a q and µ ( b , b , b ) = b r = b s . Since { p, q } ∩ { r, s } 6 = ∅ , there is an i ∈ { p, q } ∩ { r, s } , andwe have ( µ ( a , a , a ) , µ ( b , b , b )) = ( a i , b i ) ∈ ρ . (cid:3) Proposition 4.5.
The classes listed in Theorem 4.1 are ( J , SM ) -stable.Proof. Since intersections of ( J , SM )-stable classes are ( J , SM )-stable, it is enoughto prove the claim for the meet-irreducible classes. One can read off from Figure 3that the meet-irreducible classes are Ω , Ω ≤ , Ω ≥ , Ω = , , Ω = , , Ω ∗ ∪ C , Ω ∗ ∪ C , Ω ∗ ∪ C , Ω ∗ ∪ C , S − , S + , M , M , U , W , U , W , R . We can further simplify thetask with the help of Lemma 4.2, which asserts that if C is ( J , SM )-stable, then soare C , C n , and C d . Thus the only classes we need to consider are the followingeight: Ω , Ω ≤ , Ω = , , Ω ∗ ∪ C , S − , M , U , R .The classes Ω , M , and U are clones containing SM , so they are obviously ( J , SM )-stable. The same holds for the class Ω ∗ , which is also closed under left compositionwith {∧ , ∨} , so it follows by Lemma 4.3 that the class Ω ∗ ∪ C is ( J , SM )-stable.The classes Ω ≤ and Ω = , are clearly minor-closed, because for any f ∈ Ω andany σ : [ m ] → [ n ], it holds that f σ ( ) = f ( σ ) = f ( ) and f σ ( ) = f ( σ ) = f ( ).It is easy to see that they are also closed under left composition with { µ } , because µ preserves every subset of { , } by Lemma 4.4.Consider now the class S − . Let f ∈ S − and σ : [ m ] → [ n ]. Let a ∈ { , } m . Wehave f σ ( a ) = f ( a σ ) and f σ ( a ) = f ( a σ ). Since f ∈ S − , we have f ( a σ ) ∧ f ( a σ ) = 0,and since a σ = a σ , it follows that f σ ( a ) ∧ f σ ( a ) = 0, so f σ ∈ S − . Let now f, g, h ∈ S − , all n -ary, and let ϕ := µ ( f, g, h ). We need to show that ϕ ( a ) ∧ ϕ ( a ) = 0for every a ∈ { , } n . Let a ∈ { , } n . If ϕ ( a ) = 0, then ϕ ( a ) ∧ ϕ ( a ) = 0. Assumethat ϕ ( a ) = 1. Then at least two of f ( a ), g ( a ), and h ( a ) are equal to 1. Since f, g, h ∈ S − , we have f ( a ) ∧ f ( a ) = 0, g ( a ) ∧ f ( a ) = 0, h ( a ) ∧ h ( a ) = 0, soat least two of f ( a ), g ( a ), and h ( a ) are equal to 0. Consequently ϕ ( a ) = 0, so ϕ ( a ) ∧ ϕ ( a ) = 1 ∧ R . Let f ∈ R and σ : [ m ] → [ n ]. We have f σ ( a ) = f ( a σ ) = f ( a σ ) = f ( a σ ) = f σ ( a ) for every a ∈ { , } m , so f σ ∈ R . Let now f, g, h ∈ R , all n -ary, and let ϕ := µ ( f, g, h ). We have ϕ ( a ) = µ ( f ( a ) , g ( a ) , h ( a )) = µ ( f ( a ) , g ( a ) , h ( a )) = ϕ ( a ) for every a ∈ { , } n , so ϕ ∈ R . (cid:3) It remains to show that there are no further ( J , SM )-stable classes than the oneslisted in Theorem 4.1. To this end, we show that each class K is generated by anysubset F of K that is not included in any proper subclass of K , i.e., for each propersubclass C of K , the set F contains an element of K \ C . Since there are only afinite number of classes, every proper subclass of K is included in a lower cover of K , and therefore it is sufficient to consider only lower covers of K . This will beestablished in the propositions that comprise the remainder of this section.Among our main tools are stratified terms and Lemma 2.9, which provides afairly simple way to deal with classes that are clones. Lemma 2.9 is a powerful,generic tool, but, unfortunately, sometimes it may be a bit tricky to build all thestratified terms that are required for the application of the lemma. We need todevelop another tool, tailored for ( J , SM )-stability, that is easier to use in suchcases and is applicable also for classes that are not clones.For any S ⊆ { , } n , let S := { a | a ∈ S } , ↑ S := { u ∈ { , } n | ∃ a ∈ S : a ≤ u } , ↓ S := { u ∈ { , } n | ∃ a ∈ S : u ≤ a } . Lemma 4.6.
Let T and F be subsets of { , } n ( possibly empty ) such that u (cid:2) u ′ for all u , u ′ ∈ T , v (cid:2) v ′ for all v , v ′ ∈ F , and u (cid:2) v for all u ∈ T and v ∈ F .Then the following statements hold. (i) ↑ T ∩ ↓ F = ∅ , ↑ T ∩ ↓ T = ∅ and ↑ F ∩ ↓ F = ∅ . (ii) There exists an upset U of { , } n of cardinality n − such that T ⊆ U , F ⊆ U , and U ∩ U = ∅ . (iii) Consequently, there exists an n -ary function f ∈ SM such that f ( a ) = 1 for all a ∈ T and f ( b ) = 0 for all b ∈ F . AJORITY-CLOSED MINIONS OF BOOLEAN FUNCTIONS 13
Proof. (i) Suppose, to the contrary, that there is an a ∈ ↑ T ∩ ↓ F . Then there exist u ∈ T and v ∈ F such that u ≤ a ≤ v , which contradicts the assumption that u (cid:2) v . We conclude that ↑ T ∩ ↓ F = ∅ .Now suppose, to the contrary, that there is an a ∈ ↑ T ∩ ↓ T . Then there exist u ∈ T and v ∈ T such that u ≤ a ≤ v . Since v = v and v ∈ T , this contradictsour hypothesis on T . We conclude that ↑ T ∩ ↓ T = ∅ , and a similar proof showsthat ↑ F ∩ ↓ F = ∅ .(ii) Note that the cardinality of any subset S ⊆ { , } n satisfying S ∩ S = ∅ is bounded above by 2 n − . We claim that if S is an upset of { , } n such that S ∩ S = ∅ , then there exists an upset U of cardinality 2 n − such that S ⊆ U and U ∩ U = ∅ . We prove the claim by induction on the number 2 n − − | S | . Thebasis of induction is the case when | S | = 2 n − ; in this case the claim is trivial.Suppose the claim holds when | S | ≥ m ( m ≤ n − ). Consider now an upset S with | S | = m − S ∩ S = ∅ . Let w be a maximal element of { , } n \ ( S ∪ S ).We claim that S ′ := S ∪ { w } is an upset of { , } n satisfying S ′ ∩ S ′ = ∅ . Since w ∈ { , } n \ ( S ∪ S ), we also have w ∈ { , } n \ ( S ∪ S ); therefore S ′ ∩ S ′ = ( S ∪ { w } ) ∩ ( S ∪ { w } )= ( S ∩ S ) ∪ ( S ∩ { w } ) ∪ ( { w } ∩ S ) ∪ ( { w } ∩ { w } ) = ∅ . It remains to show that S ′ is an upset. Since S is an upset, we only need to verifythat u ∈ S ′ for every u ∈ { , } n such that w ≤ u . If u = w then we are done, soassume that w < u . Since w is a maximal element of { , } n \ ( S ∪ S ), it follows that u ∈ S ∪ S . Suppose, to the contrary, that u ∈ S . Since S is a downset and w < u ,it follows that w inS . But this is a contradiction because w ∈ { , } n \ ( S ∪ S ). Weconclude that u ∈ S ⊆ S ′ , as desired.We have shown that S ′ is an upset satisfying S ′ ∩ S ′ = ∅ . By the inductionhypothesis, there exists an upset U of cardinality 2 n − such that S ′ ⊆ U and U ∩ U = ∅ ; this choice is also good for the given upset S because S ⊆ S ′ .The statement follows by considering the upset S := ↑ ( T ∪ F ). By part (i), itsatisfies S ∩ S = ↑ ( T ∪ F ) ∩ ↑ ( T ∪ F ) = ( ↑ T ∪ ↑ F ) ∩ ( ↑ T ∪ ↑ F )= ( ↑ T ∪ ↑ F ) ∩ ( ↑ T ∪ ↑ F ) = ( ↑ T ∪ ↑ F ) ∩ ( ↓ T ∪ ↓ F )= ( ↑ T ∩ ↓ T ) ∪ ( ↑ T ∩ ↓ F ) ∪ ( ↑ F ∩ ↓ T ) ∪ ( ↑ F ∩ ↓ F ) = ∅ , so there exists an upset U of cardinality 2 n − such that S ⊆ U and U ∩ U = ∅ . Itclearly holds that T ⊆ S ⊆ U . Furthermore F ⊆ S ⊆ U , so F ⊆ U .(iii) Let U be the upset provided by part (ii), and define the function f : { , } n →{ , } by the rule f ( a ) = 1 if a ∈ U and f ( a ) = 0 if a ∈ U . Since U ∩ U = ∅ ,the function is well defined and self-dual. Since | U | = | U | = 2 n − , it follows that U ∪ U = { , } n ; therefore f is a total function. Since U is an upset, the functionis monotone, and since T ⊆ U and F ⊆ U , the function takes the prescribed valueson T and F . (cid:3) Definition 4.7.
Let G be a set of Boolean functions, and for each n ∈ N , denoteby G n the set of all n -ary minors of functions in G . Let f be an n -ary Booleanfunction. We say that f is G -bisectable if the following three conditions hold:(A) For all a , a ′ ∈ f − (1) there exists a τ ∈ G n such that τ ( a ) = τ ( a ′ ) = 1.(B) For all b , b ′ ∈ f − (0) there exists a τ ∈ G n such that τ ( b ) = τ ( b ′ ) = 0.(C) For all a ∈ f − (1) and for all b ∈ f − (0) there exists a τ ∈ G n such that τ ( a ) = 1 and τ ( b ) = 0.We say that a class C ⊆ Ω is G -bisectable if every function in C is G -bisectable. Lemma 4.8.
Let G ⊆ Ω and f ∈ Ω . If f is G -bisectable, then f ∈ h G i ( J , SM ) .Proof. Let f ∈ C with ar( f ) = n , and assume that f is G -bisectable. Let N bethe cardinality of G n (there are only a finite number of n -ary Boolean functions,so the set G n is certainly finite), and assume that ϕ , . . . , ϕ N is an enumerationof the functions in G n in some fixed order. Let ϕ : { , } n → { , } N , ϕ ( a ) :=( ϕ ( a ) , . . . , ϕ N ( a )). Let T := ϕ ( f − (1)) and F := ϕ ( f − (0)). Let u , v ∈ T ; thenthere exist a , a ′ ∈ f − (1) such that f ( a ) = u and f ( a ′ ) = v . By condition (A),there exists an index i ∈ [ N ] such that ϕ i ( a ) = ϕ i ( a ′ ) = 1; hence u = ϕ ( a ) (cid:2) ϕ ( a ′ ) = v . Similarly, if u , v ∈ F , then there exist b , b ′ ∈ f − (0) such that f ( b ) = u and f ( b ′ ) = v . By condition (B), there exists an index j ∈ [ N ] suchthat ϕ j ( b ) = ϕ j ( b ′ ) = 0; hence u = ϕ ( b ) (cid:2) ϕ ( b ′ ) = v . If u ∈ T and v ∈ F ,then there exist a ∈ f − (1) and b ∈ f − (0) such that f ( a ) = u and f ( b ) = v . Bycondition (C), there exists an index k ∈ [ N ] such that ϕ k ( a ) = 1 and ϕ k ( b ) = 0;hence u = ϕ ( a ) (cid:2) ϕ ( b ) = v . Therefore the sets T and F satisfy the hypothesesof Lemma 4.6, and it follows that there exists an N -ary function h ∈ SM suchthat T ⊆ h − (1) and F ⊆ h − (0). Then it holds that f = h ◦ ϕ = h ( ϕ , . . . , ϕ N ).Since h ∈ SM and ϕ , . . . , ϕ N ∈ G J , we conclude with the help of Lemma 2.4 that f ∈ SM ( G J ) = h G i ( J , SM ) . (cid:3) Lemma 4.9.
Let K be a ( J , SM ) -stable class, F ⊆ K , and G ⊆ h F i ( J , SM ) . (i) If K ⊆ h G i ( J , SM ) , then h F i ( J , SM ) = K . (ii) If K is G -bisectable, then h F i ( J , SM ) = K .Proof. (i) Since G ⊆ h F i ( J , SM ) , it follows by the general properties of closure opera-tors that h G i ( J , SM ) ⊆ hh F i ( J , SM ) i ( J , SM ) = h F i ( J , SM ) . Since F ⊆ K and K is ( J , SM )-stable, we have h F i ( J , SM ) ⊆ h K i ( J , SM ) = K . Putting these inclusions together withthe assumption K ⊆ h G i ( J , SM ) , we obtain K ⊆ h G i ( J , SM ) ⊆ h F i ( J , SM ) ⊆ K .(ii) We have K ⊆ h G i ( J , SM ) by Lemma 4.8, so the statement follows by part(i). (cid:3) We now have the tools at hand for showing that each class listed in Theorem 4.1is generated by any subset that is suggested by Figure 3. We start from the bottomof the lattice and proceed gradually upwards.
Proposition 4.10. h∅i ( J , SM ) = ∅ .Proof. Trivial. (cid:3)
Proposition 4.11. (i)
Let a ∈ { , } . For any f ∈ C a , we have h f i ( J , SM ) = C a . (ii) For any f, g ∈ C with f / ∈ C and g / ∈ C , we have h f, g i ( J , SM ) = C .Proof. (i) We have f = c ( n ) a for some n ∈ N + . We obtain all constant functionstaking value a as minors of f , so C a ⊆ h f i ( J , SM ) ⊆ C a .(ii) We have f = c ( m )1 and g = c ( n )0 for some m, n ∈ N + . By part (i), h f i ( J , SM ) = C and h g i ( J , SM ) = C , so C = C ∪ C = h g i ( J , SM ) ∪ h f i ( J , SM ) ⊆ h f, g i ( J , SM ) ⊆ C . (cid:3) Lemma 4.12. (i) If f ∈ Ω , then id is a minor of f . (ii) If f ∈ Ω , then ¬ is a minor of f . (iii) If f ∈ Ω , then is a minor of f . (iv) If f ∈ Ω , then is a minor of f .Proof. By identifying all arguments, we obtain the unary minor f ′ of f that satisfies f ′ ( a ) = f ( a, . . . , a ) for a ∈ { , } . (cid:3) AJORITY-CLOSED MINIONS OF BOOLEAN FUNCTIONS 15
Proposition 4.13.
For any f ∈ SM , we have h f i ( J , SM ) = SM .Proof. We have id ∈ h f i ( J , SM ) by Lemma 4.12(i), and consequently SM = SM J = SM ( { id } J ) = h id i ( J , SM ) ⊆ h f i ( J , SM ) ⊆ SM . (cid:3) Lemma 4.14. If f ∈ Ω \ M , then f has a ternary minor f ′ satisfying f ′ (0 , ,
0) =0 , f ′ (1 , ,
0) = 1 , f ′ (1 , ,
0) = 0 , f ′ (1 , ,
1) = 1 .Proof.
Since f ∈ Ω , we have f ( ) = 0 and f ( ) = 1. Since f / ∈ M , thereexist tuples a , b ∈ { , } n such that < a < b < and 1 = a > b = 0.Without loss of generality, we may assume that a = 1 i j + k and b = 1 i + j k for some i, j, k ∈ N + . Let f ′ be the ternary minor obtained from f by identifying three blocksof arguments: the first i arguments, the next j arguments, and the last k arguments.Then f ′ (0 , ,
0) = f ( ) = 0, f ′ (1 , ,
0) = f ( a ) = 1, f ′ (1 , ,
0) = f ( b ) = 0, and f ′ (1 , ,
1) = f ( ) = 1. (cid:3) Proposition 4.15.
For any f ∈ S \ SM , we have h f i ( J , SM ) = S .Proof. By Lemma 4.14, there is a ternary minor f ′ ≤ f such that f ′ (0 , ,
0) = 0, f ′ (1 , ,
0) = 1, f ′ (1 , ,
0) = 0, f (1 , , S is minor-closed, thefunction f ′ is self-dual, so f ′ (0 , ,
1) = 0 and f ′ (0 , ,
1) = 1. By Lemma 4.9 itsuffices to show that S is G -bisectable for G := { f ′ } . Note that id ≤ f ′ . So, let θ ∈ S . We verify that conditions (A), (B), and (C) of Definition 4.7 are satisfied.Condition (A): Let a , a ′ ∈ θ − (1). We have a = a ′ and / ∈ { a , a ′ } . Then oneof the following cases must occur, for some i , j , k : (i) (cid:0) a i a ′ i (cid:1) = (cid:0) (cid:1) , in which caseid i ( a ) = id i ( a ′ ) = 1; (ii) (cid:0) a i a j a k a ′ i a ′ j a ′ k (cid:1) = (cid:0) (cid:1) , in which case f ′ jik ( a ) = f ′ jik ( a ′ ) = 1.Condition (B): Let b , b ′ ∈ θ − (0). We have b = b ′ and / ∈ { b , b ′ } . Then oneof the following cases must occur, for some i , j , k : (i) (cid:0) b i b ′ i (cid:1) = (cid:0) (cid:1) , in which caseid i ( b ) = id i ( b ′ ) = 0; (ii) (cid:0) b i b j b k b ′ i b ′ j b ′ k (cid:1) = (cid:0) (cid:1) , in which case f ′ kij ( b ) = f ′ kij ( b ′ ) = 0.Condition (C): Let a ∈ θ − (1) and b ∈ θ − (0). We have = a = b = . Thenone of the following cases must occur, for some i , j , k : (i) (cid:0) a i b i (cid:1) = (cid:0) (cid:1) , in whichcase id i ( a ) = 1, id i ( b ) = 0; (ii) (cid:0) a i a j a k b i b j b k (cid:1) = (cid:0) (cid:1) , in which case f ′ jik ( a ) = 1, f ′ jik ( b ) = 0. (cid:3) Proposition 4.16.
For any f, g ∈ S with f / ∈ S and g / ∈ S , we have h f, g i ( J , SM ) = S .Proof. We have f ∈ S and g ∈ S . By Lemma 4.12, we have id ≤ g and ¬ ≤ f .By Lemma 4.9(i), it suffices to show that S ⊆ h id , ¬i ( J , SM ) . Since S = h µ, ¬i , thisfollows by Lemma 2.9 and Remark 2.10 from the following equivalences of terms: ¬ ( ¬ ( x )) ≡ id( x ) , ¬ ( µ ( x , x , x )) ≡ µ ( ¬ ( x ) , ¬ ( x ) , ¬ ( x )) . (cid:3) Proposition 4.17. (i)
For any f ∈ R \ C , we have h f i ( J , SM ) = R . (ii) For any f, g ∈ R ∪ C such that f / ∈ R and g / ∈ C , we have h f, g i ( J , SM ) = R ∪ C . (iii) For any f, g ∈ R such that f / ∈ R ∪ C and g / ∈ R ∪ C , we have h f, g i ( J , SM ) = R .Proof. (i) Since f ∈ Ω , we have f ( ) = f ( ) = 0. Since f / ∈ C , there exists atuple a such that f ( a ) = 1. Moreover, since f ∈ R , we have f ( a ) = 1. By identifyingarguments in a suitable way, we obtain + as a minor of f . By Lemma 4.9, it sufficesto show that R is G -bisectable for G := { + } . So, let θ ∈ R . Condition (A): Let a , a ′ ∈ θ − (1). We have { a , a ′ } ∩ { , } = ∅ . Then one ofthe following cases must occur, for some i , j : (i) (cid:0) a i a j a ′ i a ′ j (cid:1) = (cid:0) (cid:1) , in which case+ ij ( a ) = + ij ( a ′ ) = 1; (i) (cid:0) a i a j a ′ i a ′ j (cid:1) = (cid:0) (cid:1) , in which case + ij ( a ) = + ij ( a ′ ) = 1.Condition (B) holds because 0 ≤ + ∈ G .Condition (C): Let a ∈ θ − (1) and b ∈ θ − (0). We have a = b , a = b , a / ∈ { , } . Then one of the following cases must occur, for some i , j : (i) (cid:0) a i a j b i b j (cid:1) = (cid:0) (cid:1) , in which case + ij ( a ) = 1, + ij ( b ) = 0; (ii) (cid:0) a i a j b i b j (cid:1) = (cid:0) (cid:1) , in which case+ ij ( a ) = 1, + ij ( a ′ ) = 0.(ii) We have f ∈ C and g ∈ R \ C . By Lemma 4.11(i) and part (i), itholds that h f i ( J , SM ) = C and h g i ( J , SM ) = R . Therefore R ∪ C = R ∪ C = h f i ( J , SM ) ∪ h g i ( J , SM ) ⊆ h f, g i ( J , SM ) ⊆ R ∪ C .(iii) We have f ∈ R \ C and g ∈ R \ C , so by part (i) and Lemma 4.2, h g i ( J , SM ) = R and h f i ( J , SM ) = R . Therefore R = R ∪ R = h f i ( J , SM ) ∪h g i ( J , SM ) ⊆ h f, g i ( J , SM ) ⊆ R . (cid:3) Lemma 4.18. If f ∈ S − \ C , then is a minor of f .Proof. Since f ∈ Ω , we have f ( ) = f ( ) = 0. Since f / ∈ C , there is a tuple a such that f ( a ) = 1. Since f ∈ S − , we must have f ( a ) = 0. Assume, without loss ofgenerality, that a = 1 i n − i . By identifying the first i arguments and the last n − i arguments, we obtain as a minor of f . (cid:3) Proposition 4.19.
For any f ∈ ( U ∩ W ) \ C , we have h f i ( J , SM ) = U ∩ W .Proof. Let f ∈ ( U ∩ W ) \ C . By Lemma 4.18, we have ≤ f . By Lemma 4.9 itsuffices to show that U ∩ W is G -bisectable for G := { } . So, let θ ∈ U ∩ W .Condition (A): Let a , a ′ ∈ θ − (1). Since θ ∈ U , there exists an i such that a i = a ′ i = 1, and since θ ∈ W , there exists a j such that a j = a ′ j = 0, and we have ij ( a ) = ij ( b ) = 1.Condition (B) holds because 0 ≤ ∈ G .Condition (C): Let a ∈ θ − (1) and b ∈ θ − (0). We have a = b and a / ∈ { , } .Then there exist an i such that a i = b i and a j such that a j = a i . Then we have τ ( a ) = 1 and τ ( b ) = 0 for some τ ∈ { ij , ji } . (cid:3) Proposition 4.20.
For any f ∈ U \ ( U ∩ W ) , we have h f i ( J , SM ) = U .Proof. Let f ∈ U \ ( U ∩ W ). Since f ∈ Ω , we have f ( ) = f ( ) = 0. Since f / ∈ W , there exist tuples a and b such that f ( a ) = f ( b ) = 1 and a ∨ b = . Since f ∈ U , it holds that a ∧ b = and f ( a ) = f ( b ) = 0. By identifying argumentsin a suitable way, we see that f has a ternary minor f ′ satisfying f ′ (0 , ,
0) = 0, f ′ (0 , ,
1) = 0, f ′ (0 , ,
0) = 0, f ′ (1 , ,
1) = 0, f ′ (1 , ,
0) = 1, f ′ (1 , ,
1) = 1. ByLemma 4.9 it suffices to show that U is G -bisectable for G := { f ′ } . Note that = f ′ . So, let θ ∈ U .Condition (A): Let a , a ′ ∈ θ − (1). We have a ∧ a ′ = and { a , a ′ } ∩ { , } = ∅ .Then one of the following cases must occur, for some i , j , k : (i) (cid:0) a i a j a ′ i a ′ j (cid:1) = (cid:0) (cid:1) ,in which case ij ( a ) = ij ( a ′ ) = 1; (ii) (cid:0) a i a j a k a ′ i a ′ j a ′ k (cid:1) = (cid:0) (cid:1) , in which case f ′ ijk ( a ) = f ′ ijk ( a ′ ) = 1.Condition (B) holds because 0 ≤ f ′ ∈ G .Condition (C): Shown as in the proof of Proposition 4.19. (cid:3) Proposition 4.21.
For any f ∈ MU \ SM , we have h f i ( J , SM ) = MU .Proof. Let f ∈ MU \ SM . Since f ∈ Ω , we have f ( ) = 0 and f ( ) = 1. Since f / ∈ S , there exists a tuple u such that f ( u ) = f ( u ). Since u ∧ u = and f ∈ U , AJORITY-CLOSED MINIONS OF BOOLEAN FUNCTIONS 17 we must have f ( u ) = f ( u ) = 0. By identifying arguments in a suitable way, weobtain ∧ as a minor of f . Note that id ≤ ∧ .By Lemma 4.9(i), it suffices to show that MU ⊆ h id , ∧i ( J , SM ) . Since MU = h µ, ∧i , this follows by Lemma 2.9 and Remark 2.10 from the following equivalencesof terms: ∧ ( µ ( x , x , x ) , x ) ≡ µ ( ∧ ( x , x ) , ∧ ( x , x ) , ∧ ( x , x )) , ∧ ( ∧ ( x , x ) , x ) ≡ µ ( ∧ ( x , x ) , ∧ ( x , x ) , ∧ ( x , x )) . (cid:3) Proposition 4.22.
For any f, g ∈ MU with f / ∈ MU and g / ∈ C , we have h f, g i ( J , SM ) = MU .Proof. Since MU = MU ∪ C and MU ∩ C = ∅ , we have f ∈ C and g ∈ MU .Therefore 0 ≤ f and id ≤ g . By Lemma 4.9(i), it suffices to show that MU ⊆h id , i ( J , SM ) . Since MU = h µ, i , this follows by Lemma 2.9 and Remark 2.10 fromthe following equivalences of terms:0(0( x )) ≡ µ (0( x ) , x ) , x )) , µ ( x , x , x )) ≡ µ (0( x ) , x ) , x )) . (cid:3) Proposition 4.23.
For any f ∈ U \ MU , we have h f i ( J , SM ) = U .Proof. By Lemma 4.14, f has a ternary minor f ′ satisfying f ′ (0 , ,
0) = 0, f ′ (1 , ,
0) = 1, f ′ (1 , ,
0) = 0, f ′ (1 , ,
1) = 1. Since U is minor-closed, we have f ′ ∈ U , and therefore f ′ ( u ) = 0 for all u ≤ (0 , , f ′ (0 , ,
0) = 0, f ′ (0 , ,
1) = 0, and f ′ (0 , ,
1) = 0. By Lemma 4.9 it suffices to show that U is G -bisectable for G := { f ′ } . Note that id ≤ f ′ and ∧ = f ′ . So, let θ ∈ U .Condition (A): Let a , a ′ ∈ θ − (1). We have a ∧ a ′ = . Therefore id i ( a ) =id i ( a ′ ) = 1 for some i .Condition (B): Let b , b ′ ∈ θ − (0). Then b and b ′ are distinct from , so thereexist indices i and j such that b i = b ′ j = 0. Then we have ∧ ij ( b ) = ∧ ij ( b ′ ) = 0.Condition (C): Let a ∈ θ − (1) and b ∈ θ − (0). We have a = b , a = , b = .Then one of the following cases must occur, for some i , j , k : (i) (cid:0) a i b i (cid:1) = (cid:0) (cid:1) , in whichcase id i ( a ) = 1, id i ( b ) = 0; (ii) (cid:0) a i a j a k b i b j b k (cid:1) = (cid:0) (cid:1) , in which case f ′ jik ( a ) = 1, f ′ jik ( b ) = 0. (cid:3) Proposition 4.24.
For any f, g ∈ U ∪ C with f / ∈ U and g / ∈ MU , we have h f, g i ( J , SM ) = U ∪ C .Proof. Since U and C are disjoint and C ⊆ MU , we have f ∈ C and g ∈ U \ MU . By Propositions 4.11(i) and 4.23, we have h f i ( J , SM ) = C and h g i ( J , SM ) = U .Consequently, U ∪ C = h g i ( J , SM ) ∪ h f i ( J , SM ) ⊆ h f, g i ( J , SM ) ⊆ U ∪ C . (cid:3) Proposition 4.25.
For any f, g ∈ U with f / ∈ U and g / ∈ U ∪ C , we have h f, g i ( J , SM ) = U .Proof. We have f ∈ U , so id ≤ f . Since g ∈ U \ C ⊆ S − \ C , we have ≤ g by Lemma 4.18. By Lemma 4.9(i), it suffices to show that U ⊆ h id , i ( J , SM ) .Since U = h µ, i , this follows by Lemma 2.9 and Remark 2.10 from the followingequivalences of terms: ( µ ( x , x , x ) , x ) ≡ µ ( ( x , x ) , ( x , x ) , ( x , x )) , ( x , µ ( x , x , x )) ≡ µ ( ( x , x ) , ( x , x ) , ( x , x )) , ( ( x , x ) , x ) ≡ µ ( ( x , x ) , ( x , x ) , ( x , x )) , ( x , ( x , x )) ≡ µ (id( x ) , id( x ) , ( x , x )) . (cid:3) Proposition 4.26. (i)
For any f, g ∈ M with f / ∈ MU and g / ∈ MW , we have h f, g i ( J , SM ) = M . (ii) For any f, g ∈ M ∗ with f / ∈ M and g / ∈ MU , we have h f, g i ( J , SM ) = M ∗ . (iii) For any f, g, h ∈ M with f / ∈ M ∗ , g / ∈ M ∗ , and h / ∈ C , we have h f, g, h i ( J , SM ) = M .Proof. (i) Since f ∈ Ω , we have f ( ) = 0 and f ( ) = 1. Since f / ∈ U , thereexist tuples a and b such that f ( a ) = f ( b ) = 1 and a ∧ b = , i.e., b ≤ a .Since f ∈ M , we have f ( a ) = 1. By identifying arguments, we obtain ∨ as aminor of f , i.e., ∨ ∈ h f i ( J , SM ) . In a similar way we can show that ∧ ∈ h g i ( J , SM ) .Note that id is a minor of ∨ and ∧ . By Lemma 4.9(i), it suffices to show that M ⊆ h id , ∧ , ∨i ( J , SM ) . Since M = h µ, ∧ , ∨i , this follows by Lemma 2.9 andRemark 2.10 from the following equivalences of terms: ∧ ( ∧ ( x , x ) , x ) ≡ µ ( ∧ ( x , x ) , ∧ ( x , x ) , ∧ ( x , x )) , ∧ ( ∨ ( x , x ) , x ) ≡ µ ( ∧ ( x , x ) , ∧ ( x , x ) , id( x )) , ∧ ( µ ( x , x , x ) , x ) ≡ µ ( ∧ ( x , x ) , ∧ ( x , x ) , ∧ ( x , x )) , ∨ ( ∨ ( x , x ) , x ) ≡ µ ( ∨ ( x , x ) , ∨ ( x , x ) , ∨ ( x , x )) , ∨ ( ∧ ( x , x ) , x ) ≡ µ ( ∨ ( x , x ) , ∨ ( x , x ) , id( x )) , ∨ ( µ ( x , x , x ) , x ) ≡ µ ( ∨ ( x , x ) , ∨ ( x , x ) , ∨ ( x , x )) . (ii) Since f ∈ M ∗ \ M , we have f = 0. Since g / ∈ U , there exist tuples a and b such that g ( a ) = g ( b ) = 1 and a ∧ b = , i.e., b ≤ a . Since g ∈ M , we have g ( a ) = g ( ) = 1. Since g ∈ Ω ∗ , we have g ( ) = 0. It follows that id and ∨ areminors of g . Since µ (id , id ,
0) = ∧ , we have ∧ ∈ h f, g i ( J , SM ) . It now follows fromProposition 4.11(i) and part (i) that M ∗ = M ∪ C = h∧ , ∨i ( J , SM ) ∪ h i ( J , SM ) ⊆h f, g i ( J , SM ) ⊆ M ∗ .(iii) We have f = 1, g = 0, and h ∈ M , so id ≤ h . Since µ (id , id ,
0) = ∧ and µ (id , id ,
1) = ∨ , we have ∧ , ∨ ∈ h f, g, h i ( J , SM ) . It follows from Proposition 4.11(ii)and part (i) that M = M ∪ C = h∧ , ∨i ( J , SM ) ∪ h , i ( J , SM ) ⊆ h f, g, h i ( J , SM ) ⊆ M . (cid:3) Proposition 4.27. (i)
For any f, g ∈ S − with f / ∈ U and g / ∈ W , we have h f, g i ( J , SM ) = S − . (ii) For any f, g ∈ S − with f / ∈ U and g / ∈ S , we have h f, g i ( J , SM ) = S − .Proof. (i) Since f ∈ Ω , we have f ( ) = f ( ) = 0. Since f / ∈ U , there exist tuples a and b such that f ( a ) = f ( b ) = 1 and a ∧ b = , i.e., a ≤ b . Since f ∈ S − ,we have f ( a ) = f ( b ) = 0. Consequently, a = b ; therefore a < b , so a ∨ b = .Similarly, g ∈ S − \ W implies that g ( ) = g ( ) = 0 and there exist tuples c and d such that g ( c ) = g ( d ) = 1, c ∨ d = , c ∧ d = . By identifying arguments,we obtain ternary minors f ′ ≤ f and g ′ ≤ g satisfying f ′ (0 , ,
0) = 0, f ′ (1 , ,
0) =1, f ′ (0 , ,
0) = 1, f ′ (1 , ,
1) = 0, f ′ (0 , ,
1) = 0, f ′ (1 , ,
1) = 0, g ′ (0 , ,
0) = 0, g ′ (1 , ,
0) = 0, g ′ (0 , ,
0) = 0, g ′ (1 , ,
1) = 0, g ′ (0 , ,
1) = 1, g ′ (1 , ,
1) = 1. ByLemma 4.9 it suffices to show that S − is G -bisectable with G := { f ′ , g ′ } . Notethat = f ′ . So, let θ ∈ S − .Condition (A): Let a , a ′ ∈ θ − (1). We have a = a ′ and { a , a ′ } ∩ { , } = ∅ .Then one of the following cases must occur, for some i , j , k : (i) (cid:0) a i a j a ′ i a ′ j (cid:1) = (cid:0) (cid:1) ,in which case ij ( a ) = ij ( a ′ ) = 1; (ii) (cid:0) a i a j a k a ′ i a ′ j a ′ k (cid:1) = (cid:0) (cid:1) , in which case f ′ kji ( a ) = f ′ kji ( a ) = 1; (iii) (cid:0) a i a j a k a ′ i a ′ j a ′ k (cid:1) = (cid:0) (cid:1) , in which case g ′ kji ( a ) = g ′ kji ( a ) = 1.Condition (B) holds because 0 ≤ f ′ ∈ G .Condition (C): Shown as in the proof of Proposition 4.19. AJORITY-CLOSED MINIONS OF BOOLEAN FUNCTIONS 19 (ii) Since f ∈ Ω , we have f ( ) = 0 and f ( ) = 1. Since f / ∈ U , there existtuples a and b such that f ( a ) = f ( b ) = 1 and a ∧ b = . Since f ∈ S − , we musthave f ( a ) = f ( b ) = 0; hence a = b . By identifying arguments, we obtain a ternaryminor f ′ ≤ f satisfying f ′ (1 , ,
1) = 1, f ′ (1 , ,
0) = 1, f ′ (0 , ,
0) = 1, f ′ (0 , ,
0) = 0, f ′ (0 , ,
1) = 0, f ′ (1 , ,
1) = 0. Since g ∈ Ω , we have g ( ) = 0 and g ( ) = 1. Since g / ∈ S , there exists a tuple u such that g ( u ) = g ( u ) =: a . Since g ∈ S − , we musthave a = 0. By identifying arguments, we get ∧ as a minor of g . By Lemma 4.9 itsuffices to show that S − is G -bisectable with G := { f ′ , ∧} . Note that id ≤ f ′ . So,let θ ∈ S − .Condition (A): Let a , a ′ ∈ θ − (1). We have a = a ′ and a = = a ′ . Then oneof the following cases must occur, for some i , j , k : (i) (cid:0) a i a ′ i (cid:1) = (cid:0) (cid:1) , in which caseid i ( a ) = id i ( a ′ ) = 1; (ii) (cid:0) a i a j a k a ′ i a ′ j a ′ k (cid:1) = (cid:0) (cid:1) , in which case f ′ kji ( a ) = f ′ kji ( a ′ ) = 1.Condition (B): Let b , b ′ ∈ θ − (0). We have b = = b ′ . Then there exist indices i and j such that b i = b ′ j = 0. Therefore ∧ ij ( b ) = ∧ ij ( b ′ ) = 0.Condition (C): Let a ∈ θ − (1) and b ∈ θ − (0). We have = a = b = . Thenone of the following cases must occur, for some i , j , k : (i) (cid:0) a i b i (cid:1) = (cid:0) (cid:1) , in whichcase id i ( a ) = 1, id i ( b ) = 0; (ii) (cid:0) a i a j a k b i b j b k (cid:1) = (cid:0) (cid:1) , in which case f ′ kji ( a ) = 1, f ′ kji ( b ) = 0. (cid:3) Proposition 4.28.
For any f, g ∈ S − ∪ C with f / ∈ S − and g / ∈ U ∪ C , we have h f, g i ( J , SM ) = S − ∪ C .Proof. We have f = 0. Since g ∈ S − \ U , the argument in the proof of Proposi-tion 4.27(ii) shows that g has a ternary minor g ′ satisfying g ′ (1 , ,
1) = 1, g ′ (1 , ,
0) = 1, g ′ (0 , ,
0) = 1, g ′ (0 , ,
0) = 0, g ′ (0 , ,
1) = 0, g ′ (1 , ,
1) = 0. ByLemma 4.9 it suffices to show that S − ∪ C is G -bisectable with G := { , g ′ } . Notethat id ≤ g ′ . So, let θ ∈ S − ∪ C .Condition (B) holds because 0 ∈ G . Conditions (A) and (C) hold vacuously if θ ∈ C . That conditions (A) and (C) are satisfied for θ ∈ S − is shown in the sameway as in the proof of Proposition 4.27(ii). (cid:3) Proposition 4.29.
For any f, g, h ∈ S − ∗ with f / ∈ S − , g / ∈ S − ∪ C , h / ∈ U , wehave h f, g, h i ( J , SM ) = S − ∗ .Proof. We have f ∈ S − , so id ≤ f by Lemma 4.12(i), and we have g ∈ S − \ C ,so ≤ g by Lemma 4.18. Since h / ∈ U , there exist tuples a and b such that h ( a ) = h ( b ) = 1 and a ∧ b = ; moreover a and b are distinct from because h ∈ Ω ∗ . Since h ∈ S − , h ( a ) = h ( b ) = 0 and consequently a = b . Therefore h has a ternary minor h ′ satisfying h ′ (0 , ,
1) = 0, h ′ (1 , ,
1) = 0, h ′ (1 , ,
0) = 1, h ′ (0 , ,
0) = 1, h ′ (0 , ,
0) = 0. By Lemma 4.9 it suffices to show that S − ∗ is G -bisectable with G := { id , , h ′ } . Note that 0 ≤ . So, let θ ∈ S − ∗ .Condition (A): Let a , a ′ ∈ θ − (1). We have a = a ′ and a = = a ′ . Then oneof the following cases must occur, for some i , j , k : (i) (cid:0) a i a ′ i (cid:1) = (cid:0) (cid:1) , in which caseid i ( a ) = id i ( a ′ ) = 1; (ii) (cid:0) a i a j a k a ′ i a ′ j a ′ k (cid:1) = (cid:0) (cid:1) , in which case h ′ kji ( a ) = h ′ kji ( a ′ ) = 1.Condition (B) holds because 0 ≤ ∈ G .Condition (C): Let a ∈ θ − (1) and b ∈ θ − (0). We have = a = b . Thenone of the following cases must occur, for some i , j : (i) (cid:0) a i b i (cid:1) = (cid:0) (cid:1) , in whichcase id i ( a ) = 1, id i ( b ) = 0; (ii) (cid:0) a i a j b i b j (cid:1) = (cid:0) (cid:1) , in which case ji ( a ) = 1, ji ( b ) = 0. (cid:3) Proposition 4.30.
For any f, g, h ∈ S −6 = with f / ∈ S − , g / ∈ S − , h / ∈ S , we have h f, g, h i ( J , SM ) = S −6 = . Proof.
We have f ∈ S − and g ∈ S − , so ¬ ≤ f and id ≤ g . Since h / ∈ S , thereexists a tuple a such that h ( a ) = h ( a ) =: a ; since h ∈ S − , we must have a = 0; notethat a / ∈ { , } because h ( ) = h ( ). Thus h has a binary minor h ′ that satisfies h ′ (0 ,
1) = h ′ (1 ,
0) = 0. By Lemma 4.9 it suffices to show that S −6 = is G -bisectablewith G := { id , ¬ , h ′ } . So, let θ ∈ S −6 = .Condition (A): Let a , a ′ ∈ θ − (1). We have a = a ′ , so there is an i such that a i = a ′ i . Then τ ( a ) = τ ( a ′ ) = 1 for some τ ∈ { id i , ¬ i } .Condition (B): Let b , b ′ ∈ θ − (0). We have { b , b ′ } 6 = { , } . If there is an i such that b i = b ′ i , then τ ( b ) = τ ( b ′ ) = 0 for some τ ∈ { id i , ¬ i } . Otherwise thereexist j and k such that b j = b ′ k = 1, b ′ j = b k = 0, and we have h ′ jk ( b ) = h ′ jk ( b ′ ) = 0.Condition (C) Let a ∈ θ − (1) and b ∈ θ − (0). We have a = b , so there existsan i such that a i = b i . Then τ ( a ) = 1 and τ ( b ) = 0 for some τ ∈ { id i , ¬ i } . (cid:3) Proposition 4.31.
For any f, g, h ∈ S − with f / ∈ S −6 = , g / ∈ S − ∗ , h / ∈ S −∗ , we have h f, g, h i ( J , SM ) = S − .Proof. We have f ∈ S − , g ∈ S − , and h ∈ S − , so 0 ≤ f , ¬ ≤ g , and id ≤ h . ByLemma 4.9 it suffices to show that S − is G -bisectable with G := { , id , ¬} . So, let θ ∈ S − .Condition (A): Let a , a ′ ∈ θ − (1). We have a = a ′ , so there is an i such that a i = a ′ i . Then τ ( a ) = τ ( a ′ ) = 1 for some τ ∈ { id i , ¬ i } .Condition (B) holds because 0 ∈ G .Condition (C): Let a ∈ θ − (1) and b ∈ θ − (0). Then a = b , so there is an i such that a i = b i , and we have τ ( a ) = 1 and τ ( b ) = 0 for some τ ∈ { id i , ¬ i } . (cid:3) Proposition 4.32. (i)
For any f, g ∈ Ω with f / ∈ S − and g / ∈ R , we have h f, g i ( J , SM ) = Ω . (ii) For any f, g ∈ Ω ∪ C with f / ∈ Ω and g / ∈ R ∪ C , we have h f, g i ( J , SM ) = Ω ∪ C .Proof. (i) Since f ∈ Ω , we have f ( ) = f ( ) = 0; since f / ∈ S − , there existsa tuple a such that f ( a ) = f ( a ) = 1. Therefore + ≤ f . Since g ∈ Ω , wehave g ( ) = g ( ) = 0; since g / ∈ R , there exists a tuple b such that g ( b ) = g ( b ).Therefore ≤ g . By Lemma 4.9 it suffices to show that Ω is G -bisectable with G := { + , } . Note that 0 is a minor of both + and . So, let θ ∈ Ω .Condition (A): Let a , a ′ ∈ θ − (1). We have { a , a ′ } ∩ { , } = ∅ . Then one ofthe following cases must occur, for some i , j : (i) (cid:0) a i a j a ′ i a ′ j (cid:1) = (cid:0) (cid:1) , in which case ij ( a ) = ij ( a ′ ) = 1; (ii) (cid:0) a i a j a ′ i a ′ j (cid:1) = (cid:0) (cid:1) , in which case + ij ( a ) = + ij ( a ′ ) = 1.Condition (B) holds because 0 ≤ + ∈ G .Condition (C): Shown as in the proof of Proposition 4.19.(ii) We have f = 1. Since g ∈ Ω , we have g ( ) = g ( ) = 0; since g / ∈ R , thereexists a tuple a such that g ( a ) = g ( a ). Therefore ≤ g . Clearly also 0 ≤ g . Wehave µ ( , ,
1) = +, so + ∈ h , i ( J , SM ) . Part (i) and Proposition 4.11(ii) yield Ω ∪ C = h + , i ( J , SM ) ∪ h , i ( J , SM ) ⊆ h f, g i ( J , SM ) ⊆ Ω ∪ C . (cid:3) Proposition 4.33. (i)
For any f, g, h ∈ Ω with f / ∈ S − , g / ∈ S +01 , and h / ∈ M , we have h f, g, h i ( J , SM ) = Ω . (ii) For any f, g, h ∈ Ω ∪ C with f / ∈ Ω , g / ∈ S − ∪ C , and h / ∈ M ∗ , wehave h f, g, h i ( J , SM ) = Ω ∪ C . (iii) For any f, g, h ∈ Ω ∪ C with f / ∈ Ω ∪ C , g / ∈ Ω ∪ C , and h / ∈ M , wehave h f, g, h i ( J , SM ) = Ω ∪ C . AJORITY-CLOSED MINIONS OF BOOLEAN FUNCTIONS 21
Proof. (i) Since f ∈ Ω , we have f ( ) = 0 and f ( ) = 1; since f / ∈ S − , thereexists a tuple a such that f ( a ) = f ( a ) = 1. Therefore ∨ ≤ f . Since g ∈ Ω , wehave g ( ) = 0 and g ( ) = 1; since g / ∈ S + , there exists a tuple b such that g ( b ) = g ( b ) = 0. Therefore ∧ ≤ g . Since h ∈ Ω \ M , it follows from Lemma 4.14 that h has a ternary minor h ′ that satisfies h ′ (0 , ,
0) = 0, h ′ (1 , ,
0) = 1, h ′ (1 , ,
0) = 0, h ′ (1 , ,
1) = 1. By Lemma 4.9 it suffices to show that Ω is G -bisectable with G := {∨ , ∧ , h ′ } . Note that id is a minor of f , g , and h . So, let θ ∈ Ω .Condition (A): Let a , a ′ ∈ θ − (1). We have a = = a ′ , so there exist i and j such that a i = a ′ j = 1. Then ∨ ij ( a ) = ∨ ij ( a ′ ) = 1.Condition (B) Let b , b ′ ∈ θ − (0). We have b = = b ′ , so there exist i and j such that b i = b ′ j = 0. Then ∧ ij ( b ) = ∧ ij ( b ′ ) = 0.Condition (C) Let a ∈ θ − (1) and b ∈ θ − (0). We have = a = b = . Thenone of the following cases must occur, for some i , j , k : (i) (cid:0) a i b i (cid:1) = (cid:0) (cid:1) , in whichcase id i ( a ) = 1, id i ( b ) = 0; (ii) (cid:0) a i a j a k b i b j b k (cid:1) = (cid:0) (cid:1) , in which case h ′ jik ( a ) = 1, h ′ jik ( b ) = 0.(ii) We have f = 0, g ∈ Ω \ S − , h ∈ Ω \ M . By Proposition 4.11(i) andpart (i), we have Ω ∪ C = h g, h i ( J , SM ) ∪ h i ( J , SM ) ⊆ h f, g, h i ( J , SM ) ⊆ Ω ∪ C .(iii) We have f = 1 and g = 0. By Lemma 4.14, h has a ternary minor h ′ thatsatisfies h ′ (0 , ,
0) = 0, h ′ (1 , ,
0) = 1, h ′ (1 , ,
0) = 0, h ′ (1 , ,
1) = 1. By Lemma 4.9it suffices to show that Ω ∪ C is G -bisectable with G := { , , h ′ } . Note thatid ≤ h ′ . So, let θ ∈ Ω ∪ C .Conditions (A) and (B) hold because 0 , ∈ G . If θ ∈ C , then condition (C)holds vacuously. In the case when θ ∈ Ω , condition (C) can be shown to hold inthe same way as in the proof of statement (i). (cid:3) Proposition 4.34. (i)
For any f, g, h ∈ Ω ∗ with f / ∈ Ω , g / ∈ S − ∗ , h / ∈ Ω ∪ C , we have h f, g, h i ( J , SM ) = Ω ∗ . (ii) For any f, g, h ∈ Ω ∗ ∪ C with f / ∈ Ω ∗ , g / ∈ Ω ∪ C , h / ∈ Ω ∪ C , we have h f, g, h i ( J , SM ) = Ω ∗ ∪ C .Proof. (i) We have f ∈ Ω , so id ≤ f . Since g ∈ Ω ∗ , we have g ( ) = 0; since g / ∈ S − , there exists a tuple a such that g ( a ) = g ( a ) = 1. Therefore g has a binaryminor g ′ satisfying g ′ (0 ,
1) = 1, g ′ (1 ,
0) = 1, g ′ (0 ,
0) = 0. We have h ∈ Ω \ C .Thus h ( ) = h ( ) = 0 and there exists a tuple b with h ( b ) = 1. Therefore h has abinary minor h ′ satisfying h ′ (0 ,
0) = 0, h ′ (1 ,
1) = 0, h ′ (0 ,
1) = 1. By Lemma 4.9 itsuffices to show that Ω ∗ is G -bisectable with G := { id , g ′ , h ′ } . Note that 0 ≤ h ′ .So, let θ ∈ Ω ∗ .Condition (A): Let a , a ′ ∈ θ − (1). We have a = = a ′ . Then one of the followingcases must occur, for some i , j : (i) (cid:0) a i a ′ i (cid:1) = (cid:0) (cid:1) , in which case id i ( a ) = id i ( a ′ ) = 1;(ii) (cid:0) a i a j a ′ i a ′ j (cid:1) = (cid:0) (cid:1) , in which case g ′ ij ( a ) = g ′ ij ( a ′ ) = 1.Condition (B) holds because 0 ≤ h ′ ∈ G .Condition (C): Let a ∈ θ − (1) and b ∈ θ − (0). We have = a = b . Thenone of the following cases must occur, for some i , j : (i) (cid:0) a i b i (cid:1) = (cid:0) (cid:1) , in which caseid i ( a ) = 1, id i ( b ) = 0; (ii) (cid:0) a i a j b i b j (cid:1) = (cid:0) (cid:1) , in which case h ′ ij ( a ) = 1, h ′ ij ( b ) = 0.(ii) We have f = 1, and g ∈ Ω , so id ≤ g . Since h ∈ Ω \ C , h has a binaryminor h ′ satisfying h ′ (0 ,
0) = 0, h ′ (1 ,
1) = 0, h ′ (0 ,
1) = 1 (see the proof of part(i)). We have also 0 ≤ h . Note that µ ( h ′ , h ′ ,
1) = +, so + ∈ h f, g, h i ( J , SM ) . Bypart (i) and Proposition 4.11(ii), we have h , i ( J , SM ) = C and h id , + , h i ( J , SM ) = Ω ∗ .Therefore Ω ∗ ∪ C = h id , + , h i ( J , SM ) ∪ h , i ( J , SM ) ⊆ h f, g, h i ( J , SM ) ⊆ Ω ∗ ∪ C . (cid:3) Proposition 4.35. (i)
For any e, f, g, h ∈ Ω = with e / ∈ Ω , f / ∈ Ω , g / ∈ S −6 = , h / ∈ S + = , we have h e, f, g, h i ( J , SM ) = Ω = . (ii) For any f, g, h ∈ Ω = with f / ∈ Ω ∪ C , g / ∈ Ω ∪ C , h / ∈ R , we have h f, g, h i ( J , SM ) = Ω = .Proof. (i) We have e ∈ Ω , so ¬ ≤ e . We have f ∈ Ω , so id ≤ f . Since g ∈ Ω = , we have g ( ) = g ( ); since g / ∈ S − , there exists a tuple a ∈ { , } n \ { , } such that g ( a ) = g ( a ) = 1. Thus g has a binary minor g ′ satisfying g ′ (1 ,
0) =1, g ′ (0 ,
1) = 1. Similarly, we can show that h has a binary minor h ′ satisfying h ′ (1 ,
0) = 0, h ′ (0 ,
1) = 0. By Lemma 4.9 it suffices to show that Ω = is G -bisectablewith G := { id , ¬ , g ′ , h ′ } . So, let θ ∈ Ω = .Condition (A): Let a , a ′ ∈ θ − (1). We have { a , a ′ } 6 = { , } . If there is an i suchthat a i = a ′ i , then τ ( a ) = τ ( a ′ ) = 1 for some τ ∈ { id i , ¬ i } . Otherwise there exist j and k such that a j = a ′ k = 1 and a ′ j = a k = 0. Then g ′ jk ( a ) = g ′ jk ( a ′ ) = 1.Condition (B): Let b , b ′ ∈ θ − (0). We have { b , b ′ } 6 = { , } . If there is an i such that b i = b ′ i , then τ ( b ) = τ ( b ′ ) = 0 for some τ ∈ { id i , ¬ i } . Otherwise thereexist j and k such that a j = a ′ k = 1 and a ′ j = a k = 0. Then h ′ jk ( a ) = h ′ jk ( a ′ ) = 0.Condition (C): Let a ∈ θ − (1) and b ∈ θ − (0). Then a = b , so there is an i such that a i = b i , and we have τ ( a ) = 1 and τ ( b ) = 0 for some τ ∈ { id i , ¬ i } .(ii) We have f ∈ Ω \ C , so f has a binary minor f ′ satisfying f ′ (1 ,
1) = 1, f ′ (0 ,
0) = 1, f ′ (1 ,
0) = 0. Similarly, g ∈ Ω \ C , so g has a binary minor g ′ satisfying g ′ (0 ,
0) = 0, g ′ (1 ,
1) = 0, g ′ (1 ,
0) = 1. Since h ∈ Ω = \ R , there exists atuple a / ∈ { , } such that h ( a ) = h ( a ). Therefore h has a binary minor h ′ satisfying h ′ (1 ,
0) = 1, h ′ (0 ,
1) = 0. By Lemma 4.9 it suffices to show that Ω = is G -bisectablewith G := { f ′ , g ′ , h ′ } . Note that 1 ≤ f ′ and 0 ≤ g ′ . So, let θ ∈ Ω = .Conditions (A) and (B) hold because 0 ≤ g ′ ∈ G and 1 ≤ f ′ ∈ G .Condition (C): Let a ∈ θ − (1) and b ∈ θ − (0). We have a = b and { a , b } 6 = { , } . Then one of the following cases must occur, for some i , j and x, y ∈{ , } : (i) (cid:0) a i a j b i b j (cid:1) = (cid:0) x xx x (cid:1) , in which case τ ( a ) = 1 and τ ( b ) = 0 for some τ ∈{ h ′ ij , h ′ ji } ; (ii) (cid:0) a i a j b i b j (cid:1) = (cid:0) x yx y (cid:1) , in which case τ ( a ) = 1 and τ ( b ) = 0 for some τ ∈ { f ′ ij , f ′ ji , g ′ ij , g ′ ji } . (cid:3) Proposition 4.36. (i)
For any f, g, h ∈ Ω ≤ with f / ∈ Ω ∗ ∪ C , g / ∈ Ω ∗ ∪ C , h / ∈ Ω = , we have h f, g, h i ( J , SM ) = Ω ≤ . (ii) For any e, f, g, h ∈ Ω = , with e / ∈ Ω ∗ , f / ∈ Ω ∗ , g / ∈ Ω = , h / ∈ S − , we have h e, f, g, h i ( J , SM ) = Ω = , .Proof. (i) We have f ∈ Ω \ C ; hence f has a binary minor f ′ satisfying f ′ (1 ,
1) = 1, f ′ (0 ,
0) = 1, f ′ (1 ,
0) = 0. We have g ∈ Ω \ C ; hence g has a binary minor g ′ satisfying g ′ (1 ,
1) = 0, g ′ (0 ,
0) = 0, g ′ (1 ,
0) = 1. We have h ∈ Ω ; hence id ≤ h .By Lemma 4.9 it suffices to show that Ω ≤ is G -bisectable with G := { id , f ′ , g ′ } .Note that 1 ≤ f ′ and 0 ≤ g ′ . So, let θ ∈ Ω ≤ .Conditions (A) and (B) hold because 1 ≤ f ′ ∈ G and 0 ≤ g ′ ∈ G .Condition (C): Let a ∈ θ − (1) and b ∈ θ − (0). We have a = b and ( a , b ) =( , ). Then one of the following cases must occur, for some i , j : (i) (cid:0) a i b i (cid:1) = (cid:0) (cid:1) ,in which case id i ( a ) = 1, id i ( b ) = 0; (ii) (cid:0) a i a j b i b j (cid:1) = (cid:0) (cid:1) , in which case f ′ ij ( a ) = 1, f ′ ij ( b ) = 0; (iii) (cid:0) a i a j b i b j (cid:1) = (cid:0) (cid:1) , in which case g ′ ji ( a ) = 1, g ′ ji ( b ) = 0.(ii) We have e ∈ Ω , f ∈ Ω , g ∈ Ω , so ¬ ≤ e , id ≤ f , 0 ≤ g . Since h / ∈ S − ,there exists a tuple a such that h ( a ) = h ( a ) = 1; since h / ∈ Ω , we must have a / ∈ { , } . Therefore h has a binary minor h ′ satisfying h ′ (1 ,
0) = 1, h ′ (0 ,
1) = 1.By Lemma 4.9 it suffices to show that Ω = , is G -bisectable with G := { , id , ¬ , h ′ } .So, let θ ∈ Ω = , . AJORITY-CLOSED MINIONS OF BOOLEAN FUNCTIONS 23
Condition (A): Let a , a ′ ∈ θ − (1). We have { a , a ′ } 6 = { , } . If there is an i suchthat a i = a ′ i , then τ ( a ) = τ ( a ′ ) = 1 for some τ ∈ { id i , ¬ i } . Otherwise there existindices j and k such that a j = a ′ k = 1 and a ′ j = a k = 0. Then h ′ jk ( a ) = h ′ jk ( a ′ ) = 1.Condition (B) holds because 0 ∈ G .Condition (C): Let a ∈ θ − (1) and b ∈ θ − (0). Then a = b , so there is an i such that a i = b i , and we have τ ( a ) = 1 and τ ( b ) = 0 for some τ ∈ { id i , ¬ i } . (cid:3) Proposition 4.37.
For any e, f, g, h ∈ Ω with e / ∈ Ω = , , f / ∈ Ω = , , g / ∈ Ω ≥ , h / ∈ Ω ≤ , we have h e, f, g, h i ( J , SM ) = Ω .Proof. We have e ∈ Ω , f ∈ Ω , g ∈ Ω , h ∈ Ω , so 0 ≤ e , 1 ≤ f , id ≤ g , ¬ ≤ h .By Lemma 4.9 it suffices to show that Ω is G -bisectable with G := { , , id , ¬} . So,let θ ∈ Ω .Conditions (A) and (B) hold because 0 , ∈ G . Condition (C) holds because if a ∈ θ − (1) and b ∈ θ − (0), then a = b , so there is an i with a i = b i , and we have τ ( a ) = 1 and τ ( b ) = 0 for some τ ∈ { id i , ¬ i } . (cid:3) Proof of Theorem 4.1.
By Proposition 4.5, the listed classes are ( J , SM )-stable.Propositions 4.10, 4.11, 4.13, 4.15, 4.16, 4.17, 4.19, 4.20, 4.21, 4.22, 4.23, 4.24,4.25, 4.26, 4.27, 4.28, 4.29, 4.30, 4.31, 4.32, 4.33, 4.34, 4.35, 4.36, 4.37, togetherwith Lemma 4.2, show that if K is one of these classes and F is a subset of K thatis not included in any of the lower covers of K , then h F i ( J , SM ) = K . It follows thatany set F ⊆ Ω generates one of the listed classes, and there are hence no further( J , SM )-stable classes. (cid:3)
5. ( C , C ) -stable classes with SM ⊆ C Theorem 4.1 allows us to describe all ( C , C )-stable classes of Boolean functionsfor clones C and C such that C is arbitrary and SM ⊆ C . Since ( C , C )-stabilityimplies ( J , SM )-stability whenever SM ⊆ C , by Lemma 2.1, it suffices to search for( C , C )-stable classes among the ( J , SM )-stable ones. To this end, we determine, foreach ( J , SM )-stable class K , all clones C and C for which it holds that KC ⊆ K and C K ⊆ K . The results are summarized in the following theorem and Table 1. Theorem 5.1.
For each ( J , SM ) -stable class K , as determined in Theorem 4.1,there exist clones C K and C K , as prescribed in Table 1, such that for every clone C , it holds that KC ⊆ K if and only if C ⊆ C K , and CK ⊆ K if and only if C ⊆ C K . The remainder of this section is devoted to the proof of Theorem 5.1. With thehelp of the following lemma, we can reduce the number of cases to consider. Oncewe have established necessary and sufficient stability conditions for a class K , weget ones for the negation, the inner negation, and the dual of K for free. Lemma 5.2.
Let K ⊆ Ω and let C be a clone. (i) The following conditions are equivalent: (a) KC ⊆ K , (b) KC ⊆ K , (c) K n C d ⊆ K n , (d) K d C d ⊆ K d . (ii) The following conditions are equivalent: (a) CK ⊆ K , (b) CK n ⊆ K n , (c) C d K ⊆ K , (d) C d K d ⊆ K d . K KC ⊆ K CK ⊆ K K KC ⊆ K CK ⊆ K Prop.iff C ⊆ . . . iff C ⊆ . . . iff C ⊆ . . . iff C ⊆ . . . Ω Ω Ω Ω ≤ Ω M Ω ≥ Ω M Ω = , Ω U Ω = , Ω W Ω = Ω Ω Ω = Ω S Ω ∗ ∪ C Ω ∗ M Ω ∗ ∪ C Ω ∗ M Ω ∗ ∪ C Ω ∗ M Ω ∗ ∪ C Ω ∗ M Ω ∗ Ω ∗ Ω ∗ Ω ∗ Ω ∗ Ω ∗ Ω ∗ Ω ∗ Ω ∗ Ω ∗ Ω ∗ Ω ∗ Ω ∪ C Ω M Ω ∪ C Ω M Ω Ω Ω ∗ Ω Ω Ω ∗ Ω ∪ C Ω M Ω ∪ C Ω M Ω ∪ C Ω M ∗ Ω ∪ C Ω M ∗ Ω ∪ C Ω M ∗ Ω ∪ C Ω M ∗ Ω Ω Ω Ω Ω Ω S − S U S + S W S −6 = S SM S + = S SM S − ∗ S U S +1 ∗ S W S −∗ S U S + ∗ S W S − ∪ C S MU S +10 ∪ C S MW S − ∪ C S MU S +01 ∪ C S MW S − S U S +10 S W S − S U S +01 S W S − S U S +11 S W S S S S S S S S S M M M M M M M ∗ M ∗ M ∗ M ∗ M ∗ M ∗ M ∗ M ∗ M ∗ M ∗ M ∗ M ∗ M M M M M M SM SM SM SM SM SM U U U U U W U ∪ C U MU U ∪ C U MW U U U U U W MU MU MU MU MU MW MU MU MU MU MU MW W W W W W U W ∪ C W MW W ∪ C W MU W W W W W U MW MW MW MW MW MU MW MW MW MW MW MU U U U U U W W W W W W U U ∩ W SM U W ∩ U SM W R S Ω R ∪ C S M R ∪ C S M R S Ω ∗ R S Ω ∗ C Ω Ω C Ω Ω ∗ C Ω Ω ∗ ∅ Ω Ω
Table 1. ( J , SM )-stable classes and their stability under right andleft composition with clones of Boolean functions. AJORITY-CLOSED MINIONS OF BOOLEAN FUNCTIONS 25
Proof. (i) We show first that condition (a) implies (b) and (c). Assume that KC ⊆ K . In order to show that KC ⊆ K , let f ∈ K , for some f ∈ K , and g , . . . , g n ∈ C .Then f ( g , . . . , g n ) = f ( g , . . . , g n ) ∈ K because f ( g , . . . , g n ) ∈ K . In order toshow that K n C d ⊆ K n , let f n ∈ K n , for some f ∈ K , and g d1 , . . . , g d n ∈ C d ,for some g , . . . , g n ∈ C . Then f n ( g d1 , . . . , g d n ) = ( f ( g , . . . , g n )) n ∈ K n because f ( g , . . . , g n ) ∈ K .The equivalence of conditions (a), (b), (c), and (d) would follow if we showthe implications (a) ⇒ (b) ⇒ (d) ⇒ (c) ⇒ (a). We have shown above that theimplication (a) ⇒ (b) is valid. The implication (b) ⇒ (d) follows by considering K in place of K , (d) ⇒ (c) follows by considering K d in place of K and C d in placeof C , and (c) ⇒ (a) follows by considering K n in place of K and C d in place of C .(ii) We show first that condition (a) implies (b) and (c). Assume that CK ⊆ K . In order to show that CK n ⊆ K n , let f ∈ C and g n1 , . . . , g n n ∈ K n ,for some g , . . . , g n ∈ K . Then f ( g n1 , . . . , g n n ) = ( f ( g , . . . , g n )) n ∈ K n because f ( g , . . . , g n ) ∈ K . In order to show that C d K ⊆ K , let f d ∈ C d , for some f ∈ C ,and g , . . . , g n ∈ K , for some g , . . . , g n ∈ K . Then f d ( g , . . . , g n ) = f ( g , . . . , g n ) = f ( g , . . . , g n ) ∈ K because f ( g , . . . , g n ) ∈ K .The equivalence of conditions (a), (b), (c), and (d) follows by considering K , K n , or K d in place of K and C or C d in place of C . (cid:3) The following lemma is our main tool for proving the sufficiency of the conditions.It provides a sufficient condition for the intersection of two classes for which asufficient condition is known. Applying this lemma, we can proceed from the topdownwards in the lattice of ( J , SM )-stable classes. Lemma 5.3.
Let K , K ⊆ Ω , and let C and C be clones. (i) If K C ⊆ K and K C ⊆ K , then ( K ∩ K )( C ∩ C ) ⊆ K ∩ K . (ii) If C K ⊆ K and C K ⊆ K , then ( C ∩ C )( K ∩ K ) ⊆ K ∩ K .Proof. (i) Assume that K C ⊆ K and K C ⊆ K . Using the monotonicityof function class composition, we get ( K ∩ K )( C ∩ C ) ⊆ K C ⊆ K and( K ∩ K )( C ∩ C ) ⊆ K C ⊆ K ; hence ( K ∩ K )( C ∩ C ) ⊆ K ∩ K .(ii) Assume that C K ⊆ K and C K ⊆ K . Using the monotonicity offunction class composition, we get ( C ∩ C )( K ∩ K ) ⊆ C K ⊆ K and ( C ∩ C )( K ∩ K ) ⊆ C K ⊆ K ; hence ( C ∩ C )( K ∩ K ) ⊆ K ∩ K . (cid:3) The following lemma will be applied frequently without explicit mention.
Lemma 5.4.
Let C be a clone. (i) If C * Ω ∗ , then C contains or ¬ . (ii) If C * Ω , then C contains , , or ¬ . (iii) If C * M , then C contains a function f satisfying f (0 , ,
1) = 1 and f (0 , ,
1) = 0 . (iv) If C * S , then C contains a function f satisfying f (0 ,
1) = f (1 , . (v) If C * U , then C contains a function f satisfying f (0 , ,
1) = 1 and f (0 , ,
0) = 1 .Proof. (i) There exists a function g ∈ C such that g ( ) = 1. By identifying allarguments, we obtain a unary minor f of g satisfying f (0) = 1. There are twounary functions satisfying this condition, namely 1 and ¬ . Since C is minor-closed,the resulting minor is also a member of C .(ii) There exists a function g ∈ C such that g ( ) = 1 or g ( ) = 0. By identifyingall arguments, we obtain a unary minor f of g satisfying f (0) = 1 or f (1) = 0.There are three unary functions satisfying this condition: 0, 1 and ¬ . Since C isminor-closed, the resulting minor is also a member of C . (iii) There exist a function g ∈ C and tuples a and b such that a ≤ b and g ( a ) > g ( b ); hence g ( a ) = 1 and g ( b ) = 0. Without loss of generality, we mayassume that a = 0 i + j k and b = 0 i j + k for some i, j, k ∈ N + . (We must have j > a < b . We may also assume that i > k > i arguments, the next j arguments,and the last k arguments, we obtain a ternary minor f of g satisfying f (0 , ,
1) = 1and f (0 , ,
1) = 0. Since C is minor-closed, we have f ∈ C .(iv) There exist a function g ∈ C and a tuple a such that g ( a ) = g ( a ). Withoutloss of generality, we may assume that a = 0 i j for some i, j ∈ N + . (We clearly musthave i > j >
0. We may assume that both i and j are nonzero by introducinga fictitious argument if necessary.) By identifying the first i arguments and the last j arguments, we obtain a binary minor f of g satisfying g (0 ,
1) = g (1 , C is minor-closed, we have f ∈ C .(v) There exist a function g ∈ C and tuples a and b such that g ( a ) = g ( b ) = 1and a ∧ b = . Without loss of generality, we may assume that a = 0 i j k and b = 0 i j k for some i, j, k ∈ N + . By identifying the first i arguments, the next j arguments, and the last k arguments, we obtain a ternary minor f of g satisfying f (0 , ,
1) = f (0 , ,
0) = 1. Since C is minor-closed, we have f ∈ C . (cid:3) In order to prove the sufficiency of the conditions, we will make use of thenoninclusions established in the following lemma.
Lemma 5.5.
Let K be a ( J , SM ) -stable class and C a clone. (1) (a) If C * Ω and Ω = ⊆ K , then KC * Ω ≤ . (b) If C * Ω and Ω = ⊆ K , then KC * Ω = , . (c) If C * Ω ∗ and R ⊆ K , then KC * Ω ∗ ∪ C . (d) If C * Ω and R ⊆ K , then KC * Ω ∪ C . (e) If C * Ω and Ω ⊆ K , then KC * Ω ∪ C . (f) If C * S and S ⊆ K , then KC * S − . (g) If C * Ω and MU ⊆ K , then KC * Ω = . (h) If C * Ω , C ⊆ S , and SM ⊆ K , then KC * Ω ≤ . (i) If C * Ω and U ∩ W ⊆ K , then KC * Ω = . (j) If C * S , C ⊆ Ω , and S − ⊆ K , then KC * S − . (k) If C * U and SM ⊆ K , then KC * U . (l) If C * U and U ∩ W ⊆ K , then KC * U . (m) If C * S and U ∩ W ⊆ K , then KC * U ∩ W . (n) If C * M , C ⊆ S , and U ∩ W ⊆ K , then KC * U . (o) If C * S and R ⊆ K , then KC * R . (2) (p) If C * M and Ω ∪ C ⊆ K , then CK * Ω ≤ . (q) If C * U and S − ⊆ K , then CK * Ω = , . (r) If C * S and S ⊆ K , then CK * Ω = . (s) If C * M and R ∪ C ⊆ K , then CK * Ω ∗ ∪ C . (t) If C * Ω ∗ and R ⊆ K , then CK * Ω = , . (u) If C * M ∗ and Ω ∪ C ⊆ K , then CK * Ω ∪ C . (v) If C * U and U ∩ W ⊆ K , then CK * S − . (w) If C * M , C ⊆ S , and S −6 = ⊆ K , then CK * S − . (x) If C * MU and U ∪ C ⊆ K , then CK * S − ∪ C . (y) If C * Ω and SM ⊆ K , then CK * Ω . (z) If C * U and MU ⊆ K , then CK * S − .Proof. (a) Since C contains 0, 1, or ¬ , and + , ↔ , → ∈ Ω = , we have that ↔ (id ,
0) = ¬ , +(id ,
1) = ¬ , or → ( ¬ , id) = ¬ is in KC , but ¬ / ∈ Ω ≤ . AJORITY-CLOSED MINIONS OF BOOLEAN FUNCTIONS 27 (b) Since C contains 0, 1, or ¬ , and id , ¬ , ∨ ∈ Ω = , we have that ¬ (0) = 1,id(1) = 1, or ∨ (id , ¬ ) = 1 is in KC , but 1 / ∈ Ω = , .(c) Since C contains 1 or ¬ , and + ∈ R , we have that +(id ,
1) = ¬ or+(id , ¬ ) = ↔ is in KC , but ¬ , ↔ / ∈ Ω ∗ ∪ C .(d) Since C contains 0, 1, or ¬ , and + ∈ R , we have that +(id ,
0) = id,+(id ,
1) = ¬ , or +(id , ¬ ) = ↔ is in KC , but id , ¬ , ↔ / ∈ Ω ∪ C .(e) Since C contains 0, 1, or ¬ , and id , ⊕ ∈ Ω , we have that KC contains ⊕ (id , id ,
0) = +, ⊕ (id , id ,
1) = ↔ , or id( ¬ ) = ¬ , but + , ↔ , ¬ / ∈ Ω ∪ C .(f) There is f ∈ C such that f (0 ,
1) = f (1 ,
0) =: a , and we have id , ⊕ ∈ S . If a = 1, then id ( f ) = f ∈ KC , but f / ∈ S − . If a = 0, then ϕ := ⊕ (id , id , f ) ∈ KC but ϕ / ∈ S − because ϕ (0 ,
1) = ⊕ (0 , ,
0) = 1, ϕ (1 ,
0) = ⊕ (1 , ,
0) = 1.(g) Since C contains 0, 1, or ¬ , and id , ∧ ∈ MU , we have that id(0) = 0,id(1) = 1, or ∧ (id , ¬ ) = 0 is in KC , but 0 , / ∈ Ω = .(h) Since ¬ ∈ C and id ∈ SM , we have id( ¬ ) = ¬ ∈ KC , but ¬ / ∈ Ω ≤ .(i) Since C contains 0, 1, or ¬ , and → ∈ U ∩ W , we have that → (id ,
0) = id, → (1 , id) = ¬ , or → (id , ¬ ) is in KC , but id , ¬ / ∈ Ω = .(j) There is f ∈ C such that f (0 ,
1) = f (1 , f (0 ,
0) = 0, f (1 ,
1) = 1; inother words, f ∈ {∧ , ∨} . The ternary functions g and h satisfying g − (1) = { (1 , , , (0 , , , (0 , , } and h − (1) = { (1 , , , (1 , , , (0 , , } are members of S − . Hence either α := g (id , id , ∧ ) or β := h (id , id , ∧ ) is in KC , but α, β / ∈ S − because α (0 ,
1) = g (0 , ,
0) = 1, α (1 ,
0) = g (1 , ,
0) = 1, β (0 ,
1) = h (0 , ,
1) = 1, β (1 ,
0) = h (1 , ,
1) = 1.(k) The class C contains an f / ∈ U . Since id ∈ SM , we have id( f ) = f ∈ KC .(l) There is f ∈ C such that f (0 , ,
1) = f (0 , ,
0) = 1. Since ∈ U ∩ W ,we have ϕ := ( f, id ) ∈ KC , but ϕ / ∈ U because ϕ (0 , ,
1) = ( f (0 , , ,
0) = (1 ,
0) = 1, ϕ (0 , ,
0) = ( f (0 , , ,
0) = (1 ,
0) = 1, and (0 , , ∧ (0 , ,
0) = .(m) There is f ∈ C such that f (0 ,
1) = f (1 ,
0) =: a . We have ∈ U ∩ W . If a = 0, then ϕ := (id , f ) ∈ KC , but ϕ / ∈ W because ϕ (1 , ,
1) = ϕ (1 , ,
0) = 1and (1 , , ∨ (1 , ,
0) = . If a = 1, then γ := ( f , id ) ∈ KC , but γ / ∈ U because γ (0 , ,
0) = γ (1 , ,
0) = 1 and (0 , , ∧ (1 , ,
0) = . Consequently, KC * U ∩ W .(n) There is f ∈ C such that f (0 , ,
1) = 1, f (0 , ,
1) = 0, f (1 , ,
0) = 0, f (1 , ,
0) = 1. Since ∈ U ∩ W , we have δ := ( f , id ) ∈ KC , but δ / ∈ U because δ (0 , , ,
0) = δ (1 , , ,
0) = 1 and (0 , , , ∧ (1 , , ,
0) = . We concludethat ( U ∩ W ) C * U .(o) There is f ∈ C such that f (0 ,
1) = f (1 ,
0) =: a . Since + ∈ R , we have ϕ :=+(id , f ) ∈ KC , but ϕ / ∈ R because ϕ (0 ,
1) = +(0 , a ) = a , ϕ (1 ,
0) = +(1 , a ) = a .(p) There is f ∈ C such that f (0 , ,
1) = 1, f (0 , ,
1) = 0. Since 0 , , id ∈ Ω ∪ C ,we have ϕ := f (0 , id , ∈ CK , but ϕ / ∈ Ω ≤ because ϕ (0) = f (0 , ,
1) = 1, ϕ (1) = f (0 , ,
1) = 0.(q) There is f ∈ C such that f (0 , ,
1) = f (0 , ,
0) = 1. Since 0 , id , ¬ ∈ S − ,we have ϕ := f (0 , id , ¬ ) ∈ CK , but ϕ / ∈ Ω = , because ϕ (0 ,
0) = f (0 , ,
1) = 1, ϕ (1 ,
1) = f (0 , ,
0) = 1.(r) There is f ∈ C such that f (0 ,
1) = f (1 , , ¬ ∈ S , we have ϕ := f (id , ¬ ) ∈ CK , but ϕ / ∈ Ω = because ϕ (0) = f (0 ,
1) = f (1 ,
0) = ϕ (1).(s) There is f ∈ C such that f (0 , ,
1) = 1, f (0 , ,
1) = 0. Since 0 , , + ∈ R ∪ C ,we have ϕ := f (0 , + , ∈ CK , but ϕ / ∈ Ω ∗ ∪ C because ϕ (0 ,
0) = f (0 , ,
1) = 1, ϕ (0 ,
1) = f (0 , ,
1) = 0.(t) Since C contains 1 or ¬ , and 0 ∈ R , we have that 1(0) = 1 or ¬ (0) = 1 isin CK , but 1 / ∈ Ω = , .(u) We have C * M or C * Ω ∗ . If C * M , then there is f ∈ C such that f (0 , ,
1) = 1, f (0 , ,
1) = 0. Since 0 , id ∈ Ω ∪ C , we have ϕ := f (0 , id , id ) ∈ CK , but ϕ / ∈ Ω ∗ ∪ C because ϕ (1 ,
1) = f (0 , ,
1) = 0, ϕ (0 ,
1) = f (0 , ,
1) = 1. If C ⊆ Ω ∗ , then C contains 1 or ¬ . Then 1(id) = 1 or ¬ (id) = ¬ is in CK , but1 / ∈ Ω ∗ and ¬ / ∈ Ω ≤ . Consequently, CK * ( Ω ∗ ∪ C ) ∩ Ω ∗ ∩ Ω ≤ = Ω ∪ C .(v) There is f ∈ C such that f (0 , ,
1) = f (0 , ,
0) = 1. Since 0 , ∈ U ∩ W ,we have ϕ := f (0 , , ) ∈ CK , but ϕ / ∈ S − because ϕ (0 ,
1) = f (0 , ,
1) = 1, ϕ (1 ,
0) = f (0 , ,
0) = 1.(w) There is f ∈ C such that f (0 , ,
1) = 1, f (0 , ,
1) = 0, f (1 , ,
0) = 0, f (1 , ,
0) = 1. Since id , ¬ , ∧ ∈ S −6 = , we have ϕ := f (id , ∧ , ¬ ) ∈ CK , but ϕ / ∈ S − because ϕ (0 ,
1) = f (0 , ,
1) = 1, ϕ (1 ,
0) = f (1 , ,
0) = 1.(x) The class C contains either an f such that f (0 , ,
1) = 1, f (0 , ,
1) = 0or a g such that g (0 , ,
1) = g (0 , ,
0) = 1. Since 0 , id ∈ U ∪ C , we have that ϕ := f (0 , id , id ) or γ := g (0 , id , id ) is in CK , but ϕ, γ / ∈ S − ∪ C because ϕ (0 ,
1) = f (0 , ,
1) = 1, ϕ (1 ,
1) = f (0 , ,
1) = 0, γ (0 ,
1) = g (0 , ,
1) = 1, γ (1 ,
0) = g (0 , ,
0) = 1.(y) There is an f ∈ C such that f / ∈ Ω . Since id ∈ SM , it holds that f (id , . . . , id n ) = f ∈ CK .(z) There is f ∈ C such that f (0 , ,
1) = f (0 , ,
0) = 1. Since id , ∧ ∈ MU ,we have ϕ := f ( ∧ , id , id ) ∈ CK , but ϕ / ∈ S − because ϕ (0 ,
1) = f (0 , ,
1) = 1, ϕ (1 ,
0) = f (0 , ,
0) = 1. (cid:3)
Proposition 5.6.
For any clone C and for K being one of the classes Ω , Ω ∗ , Ω ∗ , Ω , M , M ∗ , M ∗ , M , U , U , MU , MU , W , W , MW , MW , S , S , and SM , the following inclusions are equivalent: C ⊆ K,KC ⊆ K, KC ⊆ K, K n C d ⊆ K n , K d C d ⊆ K d ,CK ⊆ K, CK n ⊆ K n , C d K ⊆ K, C d K d ⊆ K d . Proof.
Since the classes listed in the statement are clones, the result follows imme-diately from Lemmata 2.2 and 5.2. (cid:3)
Proposition 5.7.
Let C be a clone. (i) Ω ≤ C ⊆ Ω ≤ if and only if C ⊆ Ω . (ii) C Ω ≤ ⊆ Ω ≤ if and only if C ⊆ M . (iii) Ω = , C ⊆ Ω = , if and only if C ⊆ Ω . (iv) C Ω = , ⊆ Ω = , if and only if C ⊆ U .Proof. (i) Assume first that C ⊆ Ω . Let f ∈ Ω ≤ and g , . . . , g n ∈ C . Then g i ( ) = 0 and g i ( ) = 1 for all i ∈ [ n ], so we have f ( g , . . . , g n )( ) = f ( g ( ) , . . . , g n ( )) = f ( ) ≤ f ( )= f ( g ( ) , . . . , g n ( )) = f ( g , . . . , g n )( ) , which implies that f ( g , . . . , g n ) ∈ Ω ≤ . Therefore Ω ≤ C ⊆ Ω ≤ . Conversely, if C * Ω , then Ω ≤ C * Ω ≤ by Lemma 5.5(a).(ii) Assume first that C ⊆ M . Let f ∈ C and g , . . . , g n ∈ Ω ≤ . Then g i ( ) ≤ g i ( ) for all i ∈ [ n ], so we have f ( g , . . . , g n )( ) = f ( g ( ) , . . . , g ( )) ≤ f ( g ( ) , . . . , f ( g n ( ))) = f ( g , . . . , g n )( ) . Therefore f ( g , . . . , g n ) ∈ Ω ≤ , and we have C Ω ≤ ⊆ Ω ≤ . Conversely, if C * M , then C Ω ≤ * Ω ≤ by Lemma 5.5(p). AJORITY-CLOSED MINIONS OF BOOLEAN FUNCTIONS 29 (iii) Assume first that C ⊆ Ω . Let f ∈ Ω = , and g , . . . , g n ∈ C . Then g i ( ) = 0 and g i ( ) = 1 for all i ∈ [ n ], so we have f ( g , . . . , g n )( ) = f ( g ( ) , . . . , g n ( )) = f ( ) ,f ( g , . . . , g n )( ) = f ( g ( ) , . . . , g n ( )) = f ( ) , which implies that f ( g , . . . , g n ) ∈ Ω = , . Therefore Ω = , C ⊆ Ω = , . Conversely,if C * Ω , then Ω = , C * Ω = , by Lemma 5.5(b).(iv) Assume first that C ⊆ U . Let f ∈ C and g , . . . , g n ∈ Ω = , . Then g i ( ) ∧ g i ( ) = 0 for all i ∈ [ n ]. Therefore ( g ( ) , . . . , g n ( )) and ( g ( ) , . . . , g n ( ))cannot both be true points of f , so we have f ( g , . . . , g n )( ) ∧ f ( g , . . . , g n )( ) = f ( g ( ) , . . . , g n ( )) ∧ f ( g ( ) , . . . , g n ( )) = 0 , which means that f ( g , . . . , g n ) ∈ Ω = , . Conversely, if C * U , then we have C Ω = , * Ω = , by Lemma 5.5(q) (cid:3) Proposition 5.8.
Let C be a clone. (i) Ω = C ⊆ Ω = if and only if C ⊆ Ω . (ii) C Ω = ⊆ Ω = if and only if C ⊆ Ω . (iii) Ω = C ⊆ Ω = if and only if C ⊆ Ω . (iv) C Ω = ⊆ Ω = if and only if C ⊆ S .Proof. (i) Assume first that C ⊆ Ω . Since Ω = = Ω ≤ ∩ Ω ≥ and Ω = Ω ∩ Ω ,Lemma 5.3 and Proposition 5.7(i), together with Lemma 5.2, imply that Ω = C ⊆ Ω = . Conversely, if C * Ω , then Ω = C * Ω ≤ ⊇ Ω = by Lemma 5.5(a).(ii) For any f ∈ Ω and g , . . . , g n ∈ Ω = , it holds that f ( g , . . . , g n )( ) = f ( g ( ) , . . . , g n ( )) = f ( g ( ) , . . . , g n ( )) = f ( g , . . . , g n )( ) , so f ( g , . . . , g n ) ∈ Ω = . This shows that C Ω = ⊆ Ω = for any clone C .(iii) Assume first that C ⊆ Ω . Since Ω = = Ω = , ∩ Ω = , and Ω = Ω ∩ Ω ,Lemma 5.3 and Proposition 5.7(iii), together with Lemma 5.2, imply that Ω = C ⊆ Ω = . Conversely, if C * Ω , then Ω = C * Ω = , ⊇ Ω = by Lemma 5.5(b).(iv) Assume first that C ⊆ S . Let f ∈ C and g , . . . , g n ∈ Ω = . Then g i ( ) = g i ( )for all i ∈ [ n ], so we have f ( g , . . . , g n )( ) = f ( g ( ) , . . . , g n ( )) = f ( g ( ) , . . . , g n ( )) = f ( g , . . . , g n )( ) , which implies that f ( g , . . . , g n ) ∈ Ω = . Therefore C Ω = ⊆ Ω = . Conversely, if C * S ,then C Ω = * Ω = by Lemma 5.5(r). (cid:3) Proposition 5.9.
Let C be a clone. (i) ( Ω ∗ ∪ C ) C ⊆ Ω ∗ ∪ C if and only if C ⊆ Ω ∗ . (ii) C ( Ω ∗ ∪ C ) ⊆ Ω ∗ ∪ C if and only if C ⊆ M .Proof. (i) Assume first that C ⊆ Ω ∗ . Let f ∈ Ω ∗ ∪ C and g , . . . , g n ∈ C . If f ∈ C , then f ( g , . . . , g n ) ∈ C ⊆ Ω ∗ ∪ C . If f ∈ Ω ∗ , then f ( g , . . . , g n )( ) = f ( g ( ) , . . . , g n ( )) = f ( ) = 0, so f ( g , . . . , g n ) ∈ Ω ∗ ⊆ Ω ∗ ∪ C . Therefore( Ω ∗ ∪ C ) C ⊆ Ω ∗ ∪ C . Conversely, if C * Ω ∗ , then ( Ω ∗ ∪ C ) C * Ω ∗ ∪ C byLemma 5.5(c).(ii) Assume first that C ⊆ M . Let f ∈ C and g , . . . , g n ∈ Ω ∗ ∪ C . Observe that,for any i ∈ [ n ] and for every a ∈ { , } n , it holds that g i ( ) ≤ g i ( a ) (if g i ∈ Ω ∗ ,then g i ( ) = 0 ≤ g i ( a ); if g i ∈ C , then g i ( ) = g i ( a )). Since f is monotone, itfollows that f ( g , . . . , g n )( ) = f ( g ( ) , . . . , g n ( )) ≤ f ( g ( a ) , . . . , g n ( a )) = f ( g , . . . , g n )( a ) , for every a ∈ { , } n . If f ( g , . . . , g n )( ) = 0, then f ( g , . . . , g n ) ∈ Ω ∗ . If f ( g , . . . , g n )( ) = 1, then f ( g , . . . , g n ) = 1 ∈ C . Therefore C ( Ω ∗ ∪ C ) ⊆ Ω ∗ ∪ C .Conversely, if C * M , then C ( Ω ∗ ∪ C ) * Ω ≤ ⊇ Ω ∗ ∪ C by Lemma 5.5(p). (cid:3) Proposition 5.10.
Let C be a clone. (i) ( Ω ∪ C ) C ⊆ Ω ∪ C if and only if C ⊆ Ω . (ii) C ( Ω ∪ C ) ⊆ Ω ∪ C if and only if C ⊆ M .Proof. (i) Assume first that C ⊆ Ω . Since Ω ∪ C = ( Ω ∗ ∪ C ) ∩ ( Ω ∗ ∪ C )and Ω = Ω ∗ ∩ Ω ∗ , Lemma 5.3 and Proposition 5.9(i), together with Lemma 5.2,imply that ( Ω ∪ C ) C ⊆ Ω ∪ C . Conversely, if C * Ω , then ( Ω ∪ C ) C * Ω ∪ C by Lemma 5.5(d).(ii) Assume first that C ⊆ M . Since Ω ∪ C = ( Ω ∗ ∪ C ) ∩ ( Ω ∗ ∪ C ) and M = M ∩ M , Lemma 5.3 and Proposition 5.9(ii), together with Lemma 5.2, implythat C ( Ω ∪ C ) ⊆ Ω ∪ C . Conversely, if C * M , then C ( Ω ∪ C ) * Ω ∗ ∪ C ⊇ Ω ∪ C by Lemma 5.5(s). (cid:3) Proposition 5.11.
Let C be a clone. (i) Ω C ⊆ Ω if and only if C ⊆ Ω . (ii) C Ω ⊆ Ω if and only if C ⊆ Ω ∗ .Proof. (i) Assume first that C ⊆ Ω . Since Ω = Ω ∗ ∩ Ω ∗ and Ω = Ω ∗ ∩ Ω ∗ ,Lemma 5.3 and Proposition 5.6 imply that Ω C ⊆ Ω . Conversely, if C * Ω ,then Ω C * Ω ∪ C ⊇ Ω by Lemma 5.5(d).(ii) Assume first that C ⊆ Ω ∗ . Since Ω = Ω ∗ ∩ Ω ∗ and Ω ∗ = Ω ∗ ∩ Ω ∗ ,Lemma 5.3 and Proposition 5.6 imply that C Ω ⊆ Ω . Conversely, if C * Ω ∗ ,then C Ω * Ω = , ⊇ Ω by Lemma 5.5(t). (cid:3) Proposition 5.12.
Let C be a clone. (i) ( Ω ∪ C ) C ⊆ Ω ∪ C if and only if C ⊆ Ω . (ii) C ( Ω ∪ C ) ⊆ Ω ∪ C if and only if C ⊆ M .Proof. (i) Assume first that C ⊆ Ω . Since Ω ∪ C = ( Ω ∗ ∪ C ) ∩ ( Ω ∗ ∪ C )and Ω = Ω ∗ ∩ Ω ∗ , Lemma 5.3 and Proposition 5.9(i), together with Lemma 5.2,imply that ( Ω ∪ C ) C ⊆ Ω ∪ C . Conversely, if C * Ω , then ( Ω ∪ C ) C ⊆ Ω ∪ C by Lemma 5.5(e).(ii) Assume first that C ⊆ M . Since Ω ∪ C = ( Ω ∗ ∪ C ) ∩ ( Ω ∗ ∪ C ) and M = M ∩ M , Lemma 5.3 and Proposition 5.9(ii), together with Lemma 5.2, implythat C ( Ω ∪ C ) ⊆ Ω ∪ C . Conversely, if C * M , then C ( Ω ∪ C ) * Ω ≤ ⊇ Ω ∪ C by Lemma 5.5(p). (cid:3) Proposition 5.13.
Let C be a clone. (i) ( Ω ∪ C ) C ⊆ Ω ∪ C if and only if C ⊆ Ω . (ii) C ( Ω ∪ C ) ⊆ Ω ∪ C if and only if C ⊆ M ∗ .Proof. (i) Assume first that C ⊆ Ω . Since Ω ∪ C = Ω ∗ ∩ ( Ω ∪ C ) and Ω = Ω ∗ ∩ Ω , Lemma 5.3 and Propositions 5.6 and 5.12(i) imply that ( Ω ∪ C ) C ⊆ Ω ∪ C . Conversely, if C * Ω , then ( Ω ∪ C ) C ⊆ Ω ∪ C ⊇ Ω ∪ C by Lemma 5.5(e).(ii) Assume first that C ⊆ M ∗ . Since Ω ∪ C = Ω ∗ ∩ ( Ω ∪ C ) and M ∗ = Ω ∗ ∩ M , Lemma 5.3 and Propositions 5.6 and 5.12(ii) imply that C ( Ω ∪ C ) ⊆ Ω ∪ C .Conversely, if C * M ∗ , then C ( Ω ∪ C ) * Ω ∪ C by Lemma 5.5(u). (cid:3) Proposition 5.14.
Let C be a clone. (i) S − C ⊆ S − if and only if C ⊆ S . (ii) C S − ⊆ S − if and only if C ⊆ U . AJORITY-CLOSED MINIONS OF BOOLEAN FUNCTIONS 31
Proof. (i) Assume first that C ⊆ S . Let f ∈ S − and g , . . . , g n ∈ C . Then, forevery a ∈ { , } n , f ( g , . . . , g n )( a ) ∧ f ( g , . . . , g n )( a ) = f ( g ( a ) , . . . , g n ( a )) ∧ f ( g ( a ) , . . . , g n ( a ))= f ( g ( a ) , . . . , g n ( a )) ∧ f ( g ( a ) , . . . , g n ( a )) = 0 , where the second equality holds because each g i ∈ S , and the last equality holdsbecause f ∈ S − . This shows that f ( g , . . . , g n ) ∈ S − . Therefore S − C ⊆ S − .Conversely, if C * S , then S − C * S − by Lemma 5.5(f).(ii) Assume first that C ⊆ U . Let f ∈ C and g , . . . , g n ∈ S − . Suppose,to the contrary, that f ( g , . . . , g n )( a ) ∧ f ( g , . . . , g n )( a ) = 1 for some a . Then f ( g ( a ) , . . . , g n ( a )) = f ( g ( a ) , . . . , g n ( a )) = 1. Since f ∈ U , there exists an i ∈ [ n ]such that g i ( a ) = g i ( a ) = 1. But then g i ( a ) ∧ g i ( a ) = 1, contradicting the fact that g i ∈ S − . We conclude that f ( g , . . . , g n ) ∈ S − . Therefore C S − ⊆ S − . Conversely,if C * U , then C S − * S − by Lemma 5.5(v) (or by Lemma 5.5(q)). (cid:3) Proposition 5.15.
Let C be a clone. (i) S −6 = C ⊆ S −6 = if and only if C ⊆ S . (ii) C S −6 = ⊆ S −6 = if and only if C ⊆ SM .Proof. (i) Assume first that C ⊆ S . Since S −6 = = Ω = ∩ S − and S = Ω ∩ S ,Lemma 5.3 and Propositions 5.8(iii) and 5.14(i) imply that S −6 = C ⊆ S −6 = .Conversely, assume that C * S . Then C * S or C * Ω . If C * S , then S −6 = C * S − by Lemma 5.5(f). If C * Ω , then S −6 = C * Ω = by Lemma 5.5(g).Consequently, S −6 = C * S − ∩ Ω = = S −6 = .(ii) We have already established in Theorem 4.1 that C S −6 = ⊆ S −6 = if C ⊆ SM .Conversely, assume that C * SM . Then C * S or C * M . If C * S , then C S −6 = * Ω = by Lemma 5.5(r). Otherwise we have C * M and C ⊆ S ; in this casewe have C S −6 = * S − by Lemma 5.5(w). Consequently, C S −6 = * Ω = ∩ S − = S −6 = . (cid:3) Proposition 5.16.
Let C be a clone. (i) S − ∗ C ⊆ S − ∗ if and only if C ⊆ S . (ii) C S − ∗ ⊆ S − ∗ if and only if C ⊆ U .Proof. (i) Assume first that C ⊆ S . Since S − ∗ = Ω ∗ ∩ S − and S = Ω ∗ ∩ S ,Lemma 5.3 and Propositions 5.6 and 5.14(i) imply that S − ∗ C ⊆ S − ∗ .Conversely, assume that C * S . Then C * S or C * Ω . If C * S , then S − ∗ C * S − by Lemma 5.5(f). Otherwise C * Ω and C ⊆ S ; in this case we have S − ∗ C * Ω ≤ by Lemma 5.5(h). Consequently, S − ∗ C * S − ∩ Ω ≤ = S − ∗ .(ii) Assume first that C ⊆ U . Since S − ∗ = Ω ∗ ∩ S − and U = Ω ∗ ∩ U ,Lemma 5.3 and Propositions 5.6 and 5.14(ii) imply that C S − ∗ ⊆ S − ∗ . Conversely,if C * U , then C S − ∗ * S − ⊇ S − ∗ by Lemma 5.5(v). (cid:3) Proposition 5.17.
Let C be a clone. (i) ( S − ∪ C ) C ⊆ S − ∪ C if and only if C ⊆ S . (ii) C ( S − ∪ C ) ⊆ S − ∪ C if and only if C ⊆ MU .Proof. (i) Assume first that C ⊆ S . Since S − ∪ C = ( Ω ∪ C ) ∩ S − ∗ and S = Ω ∩ S , Lemma 5.3 and Propositions 5.13(i) and 5.16(i) imply that ( S − ∪ C ) C ⊆ S − ∪ C .Conversely, assume that C * S . Then C * S or C * Ω . If C * S , then( S − ∪ C ) C * S − by Lemma 5.5(f). Otherwise C * Ω and C ⊆ S ; in this case wehave ( S − ∪ C ) C * Ω ≤ by Lemma 5.5(h). Consequently, ( S − ∪ C ) C * S − ∩ Ω ≤ = S − ∗ ⊇ S − ∪ C . (ii) Assume first that C ⊆ MU . Since S − ∪ C = ( Ω ∪ C ) ∩ S − ∗ and MU = M ∗ ∩ U , Lemma 5.3 and Propositions 5.13(ii) and 5.16(ii) imply that ( S − ∪ C ) C ⊆ S − ∪ C . Conversely, if C * MU , then C ( S − ∪ C ) * S − ∪ C by Lemma 5.5(x). (cid:3) Proposition 5.18.
Let C be a clone. (i) S − C ⊆ S − if and only if C ⊆ S . (ii) C S − ⊆ S − if and only if C ⊆ U .Proof. (i) Assume first that C ⊆ S . Since S − = Ω ∩ S − and S = Ω ∩ S ,Lemma 5.3 and Propositions 5.6 and 5.14(i) imply that S − C ⊆ S − .Conversely, assume that C * S . Then C * S or C * Ω . If C * S , then S − C * S − by Lemma 5.5(f). Otherwise C * Ω and C ⊆ S ; in this case we have S − C * Ω ≤ by Lemma 5.5(h). Consequently, S − C * S − ∩ Ω ≤ = S − ∗ ⊇ S − .(ii) Assume first that C ⊆ U . Since S − = Ω ∩ S − and U = Ω ∩ U ,Lemma 5.3 and Propositions 5.6 and 5.14(ii) imply that C S − ⊆ S − .Conversely, assume that C * U . Then C * Ω or C * U . If C * Ω ,then C S − * Ω by Lemma 5.5(y). If C * U , then C S − * S − by Lemma 5.5(z).Consequently, C S − * Ω ∩ S − = S − . (cid:3) Proposition 5.19.
Let C be a clone. (i) S − C ⊆ S − if and only if C ⊆ S . (ii) C S − ⊆ S − if and only if C ⊆ U .Proof. (i) Assume first that C ⊆ S . Since S − = S − ∗ ∩ S −∗ and S = S ∩ S ,Lemma 5.3 and Proposition 5.16(i), together with Lemma 5.2, imply that S − C ⊆ S − .Conversely, assume that C * S . Then C * Ω or C * S . If C * Ω , then S − C * Ω = by Lemma 5.5(i). Otherwise we have C * S and C ⊆ Ω ; in this casewe have S − C * S − by Lemma 5.5(j). Consequently, S − C * Ω = ∩ S − = S − .(ii) Assume first that C ⊆ U . Since S − = S − ∗ ∩ S −∗ and U = U ∩ U ,Lemma 5.3 and Proposition 5.16(ii), together with Lemma 5.2, imply that C S − ⊆ S − . Conversely, if C * U , then C S − * S − ⊇ S − by Lemma 5.5(v). (cid:3) Proposition 5.20.
Let C be a clone. (i) ( U ∪ C ) C ⊆ U ∪ C if and only if C ⊆ U . (ii) C ( U ∪ C ) ⊆ U ∪ C if and only if C ⊆ MU .Proof. (i) Assume first that C ⊆ U . Since U ∪ C = U ∩ ( Ω ∪ C ) and U = U ∩ Ω , Lemma 5.3 and Propositions 5.6 and 5.13(i) imply that ( U ∪ C ) C ⊆ U ∪ C .Conversely, assume that C * U . Then C * Ω or C * U . If C * Ω ,then ( U ∪ C ) C * Ω = by Lemma 5.5(g). If C * U , then ( U ∪ C ) C * U byLemma 5.5(k). Consequently, ( U ∪ C ) C * Ω = ∩ U = U .(ii) Assume first that C ⊆ MU . Since U ∪ C = U ∩ ( Ω ∪ C ) and MU = U ∩ M ∗ , Lemma 5.3 and Propositions 5.6 and 5.13(ii) imply that C ( U ∪ C ) ⊆ U ∪ C . Conversely, if C * MU , then C ( U ∪ C ) * S − ∪ C ⊇ U ∪ C byLemma 5.5(x). (cid:3) Proposition 5.21.
Let C be a clone. (i) U C ⊆ U if and only if C ⊆ U . (ii) C U ⊆ U if and only if C ⊆ U .Proof. (i) Assume first that C ⊆ U . Since U = U ∩ Ω and U = U ∩ Ω ,Lemma 5.3 and Propositions 5.6 and 5.11(i) imply that U C ⊆ U . AJORITY-CLOSED MINIONS OF BOOLEAN FUNCTIONS 33
Conversely, assume that C * U . Then C * Ω or C * U . If C * Ω ,then U C * Ω = by Lemma 5.5(i). If C * U , then U C * U by Lemma 5.5(l).Consequently, U C * Ω = ∩ U = U .(ii) Assume first that C ⊆ U . Since U = U ∩ Ω and U = U ∩ Ω ∗ ,Lemma 5.3 and Propositions 5.6 and 5.11(ii) imply that C U ⊆ U . Conversely,if C * U , then C U * S − ⊇ U by Lemma 5.5(v). (cid:3) Proposition 5.22.
Let C be a clone. (i) ( U ∩ W ) C ⊆ U ∩ W if and only if C ⊆ SM . (ii) C ( U ∩ W ) ⊆ U ∩ W if and only if C ⊆ U .Proof. (i) Assume first that C ⊆ SM . Since U ∩ W = U ∩ W and SM = U ∩ W ,Lemma 5.3 and Proposition 5.6 imply that ( U ∩ W ) C ⊆ U ∩ W .Conversely, assume that C * SM . Then C * S or C * M . If C * S , then( U ∩ W ) C * U ∩ W by Lemma 5.5(m). Otherwise we have C * M and C ⊆ S ; inthis case we have ( U ∩ W ) C * U by Lemma 5.5(n). Consequently, ( U ∩ W ) C * ( U ∩ W ) ∩ U = U ∩ W .(ii) Assume first that C ⊆ U . Since U ∩ W = U ∩ W and U = U ∩ U ,Lemma 5.3 and Proposition 5.6 imply that C ( U ∩ W ) ⊆ U ∩ W . Conversely, if C * U , then C ( U ∩ W ) * S − ⊇ U ∩ W by Lemma 5.5(v). (cid:3) Proposition 5.23.
Let C be a clone. (i) R C ⊆ R if and only if C ⊆ S . (ii) C R ⊆ R for every clone C .Proof. (i) Assume first that C ⊆ S . Let f ∈ R ( n ) and g , . . . , g n ∈ C ( m ) . It holdsthat, for all a ∈ { , } m , f ( g , . . . , g n )( a ) = f ( g ( a ) , . . . , g n ( a )) = f ( g ( a ) , . . . , g n ( a ))= f ( g ( a ) , . . . , g n ( a )) = f ( g , . . . , g n )( a ) , where the second equality holds because g i ∈ S and the third equality holds because f ∈ R . Thus f ( g , . . . , g n ) ∈ R . Therefore R C ⊆ R . Conversely, if C * S , then R C * R by Lemma 5.5(o).(ii) For any f ∈ Ω ( n ) and g , . . . , g n ∈ R ( m ) it holds that f ( g , . . . , g n )( a ) = f ( g ( a ) , . . . , g n ( a )) = f ( g ( a ) , . . . , g n ( a )) = f ( g , . . . , g n )( a ) , for all a ∈ { , } m . Therefore C R ⊆ R for any clone C . (cid:3) Proposition 5.24.
Let C be a clone. (i) ( R ∪ C ) C ⊆ R ∪ C if and only if C ⊆ S . (ii) C ( R ∪ C ) ⊆ R ∪ C if and only if C ⊆ M .Proof. (i) Assume first that C ⊆ S . Since R ∪ C = ( Ω ∪ C ) ∩ R and S = Ω ∩ S ,Lemma 5.3 and Propositions 5.10(i) and 5.23(i) imply that ( R ∪ C ) C ⊆ R ∪ C .Conversely, assume that C * S . Then C * Ω or C * S . If C * Ω ,then ( R ∪ C ) C * Ω ∪ C by Lemma 5.5(d). If C * S , then ( R ∪ C ) C * R byLemma 5.5(o). Consequently, ( R ∪ C ) C * ( Ω ∪ C ) ∩ R = R ∪ C .(ii) Assume first that C ⊆ M . Since R ∪ C = ( Ω ∪ C ) ∩ R and M = M ∩ Ω ,Lemma 5.3 and Propositions 5.10(ii) and 5.23(ii) imply that C ( R ∪ C ) ⊆ R ∪ C .Conversely, if C * M , then C ( R ∪ C ) * Ω ∗ ∪ C ⊇ R ∪ C by Lemma 5.5(s). (cid:3) Proposition 5.25.
Let C be a clone. (i) R C ⊆ R if and only if C ⊆ S . (ii) C R ⊆ R if and only if C ⊆ Ω ∗ . Proof. (i) Assume first that C ⊆ S . Since R = Ω ∩ R and S = Ω ∩ S ,Lemma 5.3 and Propositions 5.11(i) and 5.23(i) imply that R C ⊆ R .Conversely, assume that C * S . Then C * Ω or C * S . If C * Ω , then R C * Ω ∪ C by Lemma 5.5(d). If C * S , then R C * R by Lemma 5.5(o).Consequently, R C * ( Ω ∪ C ) ∩ R = R ∪ C ⊇ R .(ii) Assume first that C ⊆ Ω ∗ . Since R = Ω ∩ R and Ω ∗ = Ω ∗ ∩ Ω ,Lemma 5.3 and Propositions 5.11(ii) and 5.23(ii) imply that C R ⊆ R . Con-versely, if C * Ω ∗ , then C R * Ω = , ⊇ R by Lemma 5.5(t). (cid:3) Proposition 5.26.
Let C be a clone. (i) C C ⊆ C for every clone C . (ii) C C ⊆ C for every clone C . (iii) C C ⊆ C for every clone C . (iv) C C ⊆ C if and only if C ⊆ Ω ∗ . (v) C C ⊆ C for every clone C . (vi) C C ⊆ C if and only if C ⊆ Ω ∗ .Proof. Straightforward verification. (cid:3)
Proposition 5.27. ∅ C ⊆ ∅ and C ∅ ⊆ ∅ for every clone C .Proof. Trivial. (cid:3)
Proof of Theorem 5.1.
The theorem puts together Lemma 5.2 and Propositions 5.6,5.7, 5.8, 5.9, 5.10, 5.11, 5.12, 5.13, 5.14, 5.15, 5.16, 5.17, 5.18, 5.19, 5.20, 5.21, 5.22,5.23, 5.24, 5.25, 5.26, 5.27. (cid:3)
With the help of Post’s lattice and by reading off from Table 1, we can determinefor any pair ( C , C ) of clones which ( J , SM )-stable classes are ( C , C )-stable. If SM ⊆ C , then any ( C , C )-stable class is ( J , SM )-stable by Lemma 2.1. Therefore,in the case when SM ⊆ C , the ( C , C )-stable classes occur among the ( J , SM )-stable ones and they can be easily picked out from Table 1. In particular, we havean explicit description of ( J , C )-stable classes (“clonoids” of Aichinger and Mayr [1])and ( C, C )-stable classes for SM ⊆ C . Corollary 5.28. (a)
The ( J , MU ) -stable classes are Ω , Ω ≤ , Ω ≥ , Ω = , , Ω = , Ω ∗ ∪ C , Ω ∗ ∪ C , Ω ∗ ∪ C , Ω ∗ ∪ C , Ω ∗ , Ω ∗ , Ω ∗ , Ω ∗ , Ω ∪ C , Ω ∪ C , Ω , Ω , Ω ∪ C , Ω ∪ C , Ω ∪ C , Ω ∪ C , Ω ∪ C , Ω ∪ C , Ω , Ω , S − , S − ∗ , S −∗ , S − ∪ C , S − ∪ C , S − , S − , S − , M , M , M ∗ , M ∗ , M ∗ , M ∗ , M , M , U , U ∪ C , U , MU , MU , W , W ∪ C , W , MW , MW , U , W , U ∩ W , R , R ∪ C , R ∪ C , R , R , C , C , C , ∅ . (b) The ( J , MU ) -stable classes are Ω , Ω ≤ , Ω ≥ , Ω = , , Ω = , Ω ∗ ∪ C , Ω ∗ ∪ C , Ω ∗ ∪ C , Ω ∗ ∪ C , Ω ∗ , Ω ∗ , Ω ∪ C , Ω ∪ C , Ω , Ω ∪ C , Ω ∪ C , Ω ∪ C , Ω ∪ C , S − , S − ∗ , S −∗ , S − ∪ C , S − ∪ C , S − , M , M , M ∗ , M ∗ , U , U ∪ C , MU , W , W ∪ C , MW , U , W , U ∩ W , R , R ∪ C , R ∪ C , R , C , C , ∅ . (c) The ( J , U ) -stable classes are Ω , Ω = , , Ω = , Ω ∗ , Ω ∗ , Ω ∗ , Ω ∗ , Ω , Ω , Ω , Ω , S − , S − ∗ , S −∗ , S − , S − , S − , U , U , W , W , U , W , U ∩ W , R , R , R , C , C , C , ∅ . (d) The ( J , U ) -stable classes are Ω , Ω = , , Ω = , Ω ∗ , Ω ∗ , Ω , S − , S − ∗ , S −∗ , S − , U , W , U , W , U ∩ W , R , R , C , C , ∅ . (e) The ( J , MW ) -stable classes are Ω , Ω ≤ , Ω ≥ , Ω = , , Ω = , Ω ∗ ∪ C , Ω ∗ ∪ C , Ω ∗ ∪ C , Ω ∗ ∪ C , Ω ∗ , Ω ∗ , Ω ∗ , Ω ∗ , Ω ∪ C , Ω ∪ C , Ω , Ω , Ω ∪ C , Ω ∪ C , Ω ∪ C , Ω ∪ C , Ω ∪ C , Ω ∪ C , Ω , Ω , S + , S +1 ∗ , S + ∗ , AJORITY-CLOSED MINIONS OF BOOLEAN FUNCTIONS 35 S +10 ∪ C , S +01 ∪ C , S +10 , S +01 , S +11 , M , M , M ∗ , M ∗ , M ∗ , M ∗ , M , M , U , U ∪ C , U , MU , MU , W , W ∪ C , W , MW , MW , U , W , W ∩ U , R , R ∪ C , R ∪ C , R , R , C , C , C , ∅ . (f) The ( J , MW ) -stable classes are Ω , Ω ≤ , Ω ≥ , Ω = , , Ω = , Ω ∗ ∪ C , Ω ∗ ∪ C , Ω ∗ ∪ C , Ω ∗ ∪ C , Ω ∗ , Ω ∗ , Ω ∪ C , Ω ∪ C , Ω , Ω ∪ C , Ω ∪ C , Ω ∪ C , Ω ∪ C , S + , S +1 ∗ , S + ∗ , S +10 ∪ C , S +01 ∪ C , S +11 , M , M , M ∗ , M ∗ , U , U ∪ C , MU , W , W ∪ C , MW , U , W , W ∩ U , R , R ∪ C , R ∪ C , R , C , C , ∅ . (g) The ( J , W ) -stable classes are Ω , Ω = , , Ω = , Ω ∗ , Ω ∗ , Ω ∗ , Ω ∗ , Ω , Ω , Ω , Ω , S + , S +1 ∗ , S + ∗ , S +10 , S +01 , S +11 , U , U , W , W , U , W , W ∩ U , R , R , R , C , C , C , ∅ . (h) The ( J , W ) -stable classes are Ω , Ω = , , Ω = , Ω ∗ , Ω ∗ , Ω , S + , S +1 ∗ , S + ∗ , S +11 , U , W , U , W , W ∩ U , R , R , C , C , ∅ . (i) The ( J , S ) -stable classes are Ω , Ω = , Ω = , Ω ∗ , Ω ∗ , Ω ∗ , Ω ∗ , Ω , Ω , Ω , Ω , S , S , S , R , R , R , C , C , C , ∅ . (j) The ( J , S ) -stable classes are Ω , Ω = , Ω = , S , R , C , ∅ . (k) The ( J , M ) -stable classes are Ω , Ω ≥ , Ω ≤ , Ω = , Ω ∗ ∪ C , Ω ∗ ∪ C , Ω ∗ ∪ C , Ω ∗ ∪ C , Ω ∗ , Ω ∗ , Ω ∗ , Ω ∗ , Ω ∪ C , Ω ∪ C , Ω , Ω , Ω ∪ C , Ω ∪ C , Ω ∪ C , Ω ∪ C , Ω ∪ C , Ω ∪ C , Ω , Ω , M , M , M ∗ , M ∗ , M ∗ , M ∗ , M , M , R , R ∪ C , R ∪ C , R , R , C , C , C , ∅ . (l) The ( J , M ∗ ) -stable classes are Ω , Ω ≥ , Ω ≤ , Ω = , Ω ∗ ∪ C , Ω ∗ ∪ C , Ω ∗ ∪ C , Ω ∗ ∪ C , Ω ∗ , Ω ∗ , Ω ∪ C , Ω ∪ C , Ω , Ω ∪ C , Ω ∪ C , Ω ∪ C , Ω ∪ C , M , M , M ∗ , M ∗ , R , R ∪ C , R ∪ C , R , C , C , ∅ . (m) The ( J , M ∗ ) -stable classes are Ω , Ω ≥ , Ω ≤ , Ω = , Ω ∗ ∪ C , Ω ∗ ∪ C , Ω ∗ ∪ C , Ω ∗ ∪ C , Ω ∗ , Ω ∗ , Ω ∪ C , Ω ∪ C , Ω , Ω ∪ C , Ω ∪ C , Ω ∪ C , Ω ∪ C , M , M , M ∗ , M ∗ , R , R ∪ C , R ∪ C , R , C , C , ∅ . (n) The ( J , M ) -stable classes are Ω , Ω ≥ , Ω ≤ , Ω = , Ω ∗ ∪ C , Ω ∗ ∪ C , Ω ∗ ∪ C , Ω ∗ ∪ C , Ω ∪ C , Ω ∪ C , Ω ∪ C , Ω ∪ C , M , M , R , R ∪ C , R ∪ C , C , ∅ . (o) The ( J , Ω ) -stable classes are Ω , Ω = , Ω ∗ , Ω ∗ , Ω ∗ , Ω ∗ , Ω , Ω , Ω , Ω , R , R , R , C , C , C , ∅ . (p) The ( J , Ω ∗ ) -stable classes are Ω , Ω = , Ω ∗ , Ω ∗ , Ω , R , R , C , C , ∅ . (q) The ( J , Ω ∗ ) -stable classes are Ω , Ω = , Ω ∗ , Ω ∗ , Ω , R , R , C , C , ∅ . (r) The ( J , Ω ) -stable classes are Ω , Ω = , R , C , ∅ . Corollary 5.29. (a)
The SM -stable classes are precisely the 93 ( J , SM ) -stable classes. (b) The MU -stable classes are Ω , Ω ≤ , Ω ≥ , Ω = , , Ω = , Ω ∗ ∪ C , Ω ∗ ∪ C , Ω ∗ ∪ C , Ω ∗ ∪ C , Ω ∗ , Ω ∗ , Ω ∗ , Ω ∗ , Ω ∪ C , Ω ∪ C , Ω , Ω , Ω ∪ C , Ω ∪ C , Ω ∪ C , Ω ∪ C , Ω ∪ C , Ω ∪ C , Ω , Ω , M , M , M ∗ , M ∗ , M ∗ , M ∗ , M , M , U , U ∪ C , U , MU , MU , U , C , C , C , ∅ . (c) The MU -stable classes are Ω , Ω ∗ ∪ C , Ω ∗ ∪ C , Ω ∗ , M , M , M ∗ , U , MU , C , C , ∅ . (d) The U -stable classes are Ω , Ω = , , Ω = , Ω ∗ , Ω ∗ , Ω ∗ , Ω ∗ , Ω , Ω , Ω , Ω , U , U , U , C , C , C , ∅ . (e) The U -stable classes are Ω , Ω ∗ , U , C , C , ∅ . (f) The MW -stable classes are Ω , Ω ≤ , Ω ≥ , Ω = , , Ω = , Ω ∗ ∪ C , Ω ∗ ∪ C , Ω ∗ ∪ C , Ω ∗ ∪ C , Ω ∗ , Ω ∗ , Ω ∗ , Ω ∗ , Ω ∪ C , Ω ∪ C , Ω , Ω , Ω ∪ C , Ω ∪ C , Ω ∪ C , Ω ∪ C , Ω ∪ C , Ω ∪ C , Ω , Ω , M , M , M ∗ , M ∗ , M ∗ , M ∗ , M , M , W , W ∪ C , W , MW , MW , W , C , C , C , ∅ . (g) The MW -stable classes are Ω , Ω ∗ ∪ C , Ω ∗ ∪ C , Ω ∗ , M , M , M ∗ , W , MW , C , C , ∅ . (h) The W -stable classes are Ω , Ω = , , Ω = , Ω ∗ , Ω ∗ , Ω ∗ , Ω ∗ , Ω , Ω , Ω , Ω , W , W , W , C , C , C , ∅ . (i) The W -stable classes are Ω , Ω ∗ , W , C , C , ∅ . (j) The S -stable classes are Ω , Ω = , Ω = , Ω ∗ , Ω ∗ , Ω ∗ , Ω ∗ , Ω , Ω , Ω , Ω , S , S , S , R , R , R , C , C , C , ∅ . (k) The S -stable classes are Ω , S , R , C , ∅ . (l) The M -stable classes are Ω , Ω ≥ , Ω ≤ , Ω = , Ω ∗ ∪ C , Ω ∗ ∪ C , Ω ∗ ∪ C , Ω ∗ ∪ C , Ω ∗ , Ω ∗ , Ω ∗ , Ω ∗ , Ω ∪ C , Ω ∪ C , Ω , Ω , Ω ∪ C , Ω ∪ C , Ω ∪ C , Ω ∪ C , Ω ∪ C , Ω ∪ C , Ω , Ω , M , M , M ∗ , M ∗ , M ∗ , M ∗ , M , M , C , C , C , ∅ . (m) The M ∗ -stable classes are Ω , Ω ∗ ∪ C , Ω ∗ ∪ C , Ω ∗ , M , M , M ∗ , C , C , ∅ . (n) The M ∗ -stable classes are Ω , Ω ∗ ∪ C , Ω ∗ ∪ C , Ω ∗ , M , M , M ∗ , C , C , ∅ . (o) The M -stable classes are Ω , M , M , C , ∅ . (p) The Ω -stable classes are Ω , Ω = , Ω ∗ , Ω ∗ , Ω ∗ , Ω ∗ , Ω , Ω , Ω , Ω , C , C , C , ∅ . (q) The Ω ∗ -stable classes are Ω , Ω ∗ , C , C , ∅ . (r) The Ω ∗ -stable classes are Ω , Ω ∗ , C , C , ∅ . (s) The Ω -stable classes are Ω , C , ∅ . It is noteworthy that ( J , SM )-stability is equivalent to SM -stability. This is ex-plained by the following proposition, which also provides other equivalences forclones containing µ . Proposition 5.30. (i)
For any f ∈ Ω , we have f ∗ µ = µ ( f σ , f σ , f σ ) , where, for i ∈ [3] , σ i : [ n ] → [ n + 2] , i , j j + 2 for ≤ j ≤ n . (ii) Let G ⊆ Ω , let C := h G ∪ { µ }i , C ′ := h G i , and let C be a clone containing µ . Then a class F ⊆ Ω is ( C , C ) -stable if and only if it is ( C ′ , C ) -stable. (iii) The following are equivalent for a class F ⊆ Ω . (a) F is SM -stable. (b) F is ( J , SM ) -stable. (c) F is minor-closed and µ ( f, g, h ) ∈ F whenever f, g, h ∈ F .Proof. (i) It is easy to verify that f ( µ ( a , a , a ) , a , . . . , a n +2 )= µ ( f ( a , a , . . . , a n +2 ) , f ( a , a , . . . , a n +2 ) , f ( a , a , . . . , a n +2 ))holds for all a , . . . , a n +2 ∈ { , } . For, if a = a = a , then both left and rightsides are equal to f ( a , a , . . . , a n +2 ). If a , a , and a are not all equal, then two ofthem must be equal while the third is distinct from the other two. Assume, withoutloss of generality, that a = a = a . Then µ ( a , a , a ) = a , so the left side equals f ( a , a , . . . , a n +2 ). Since f ( a , a , a , . . . , a n +2 ) = f ( a , a , a , . . . , a n +2 ), we seethat also the right side equals f ( a , a , . . . , a n +2 ).(ii) Since C ′ ⊆ C , stability under right composition with C implies stabil-ity under right composition with C ′ . Assume now that F is ( C ′ , C )-stable. ByLemma 2.5, F is minor-closed and f ∗ g ∈ F whenever f ∈ F and g ∈ G . Moreover, f ∗ µ = µ ( f σ , f σ , f σ ), where f σ , f σ , f σ are the minors of f specified in part (i).Since F is minor-closed, we have f σ , f σ , f σ ∈ F . By our assumption, µ ∈ C , andsince F is stable under left composition with C , it follows that µ ( f σ , f σ , f σ ) ∈ F .It follows from Lemma 2.5 that F is stable under right composition with C .(iii) This is a consequence of part (ii) and Lemma 2.6. (cid:3) AJORITY-CLOSED MINIONS OF BOOLEAN FUNCTIONS 37 Concluding remarks
The results of this paper cover only a part of the clones C of Boolean functionsfor which there are only a finite number of ( J , C )-stable classes (see Theorem 1.1(i)).The description of ( J , C )-stable classes for the other clones C containing a near-unanimity operation (i.e., the clones U k , U k , MU k , MU k , W k , W k , MW k , MW k for k ≥
3) remains a topic for further research.
Acknowledgments
The author would like to thank Miguel Couceiro and Sebastian Kreinecker forinspiring discussions.This work is funded by National Funds through the FCT – Funda¸c˜ao para aCiˆencia e a Tecnologia, I.P., under the scope of the project UIDB/00297/2020(Center for Mathematics and Applications).
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