Solution sets of systems of equations over finite lattices and semilattices
SSOLUTION SETS OF SYSTEMS OF EQUATIONS OVER FINITELATTICES AND SEMILATTICES
ENDRE T ´OTH AND TAM ´AS WALDHAUSER
Abstract.
Solution sets of systems of homogeneous linear equations over fieldsare characterized as being subspaces, i.e., sets that are closed under linear com-binations. Our goal is to characterize solution sets of systems of equations overarbitrary finite algebras by a similar closure condition. We show that solutionsets are always closed under the centralizer of the clone of term operations of thegiven algebra; moreover, the centralizer is the only clone that could characterizesolution sets. If every centralizer-closed set is the set of all solutions of a system ofequations over a finite algebra, then we say that the algebra has Property (SDC).Our main result is the description of finite lattices and semilattices with Prop-erty (SDC): we prove that a finite lattice has Property (SDC) if and only if it isa Boolean lattice, and a finite semilattice has Property (SDC) if and only if it isdistributive. Introduction
In universal algebra, investigations of systems of equations usually focus on eitherfinding a solution, the complexity of finding a solution or deciding if there is a solutionat all. For us the main interest is the “shape” of the solution sets, just like in thefollowing basic result of linear algebra: solution sets of systems of homogeneous linearequations in n variables over a field K are precisely the subspaces of the vector space K n , i.e., sets of n -tuples that are closed under linear combinations. Our goal is togive a similar characterization (i.e., a kind of closure condition) for solution sets ofsystems of equations over arbitrary finite algebras.Let us fix a nonempty set A and a set F of operations on A ; then we obtain thealgebra A = ( A, F ). Any equation over A is of the form f ( x , . . . , x n ) = g ( x , . . . , x n ),where f and g are n -ary term functions. We can also say that f and g are from the set[ F ] of operations generated by F by means of compositions. After this observation wecan see that in every equation, the operations on both sides are from C := [ F ], whichwe will call the clone generated by F (Definition 2.1). We will investigate solution setsof systems of equations over finite algebras in this view. The algebraic sets studiedby B. I. Plotkin in his universal algebraic geometry [10] are essentially the same asour solution sets; the only difference being that we consider only finite systems ofequations. Recently A. Di Nola, G. Lenzi and G. Vitale characterized the solutionsets of certain systems of equations over lattice ordered abelian groups (see [3]).In our previous paper [12] we proved that for any system of equations over a clone C ,the solution set is closed under the centralizer of the clone C (see Definition 2.2). Wealso proved that for clones of Boolean functions this condition is sufficient as well. Wewill say that a clone (or the associated algebra) has Property (SDC) if closure underthe centralizer characterizes the solution sets (here SDC stands for “Solution sets areDefinable by closure under the Centralizer”). Thus clones of Boolean functions (i.e.,two-element algebras) always have Property (SDC), and in [12] we gave an example ofa three-element algebra that does not have Property (SDC). In this paper we describeall finite lattices and semilattices with Property (SDC). In Section 2 we presentthe necessary notations and definitions. In Section 3 we give a connection betweenProperty (SDC) and quantifier elimination of certain primitive positive formulas. Also Mathematics Subject Classification.
Key words and phrases. system of equations, solution set, clone, lattice, semilattice, distributivelattice, distributive semilattice, Boolean lattice, primitive positive formula, quantifier elimination. a r X i v : . [ m a t h . R A ] J u l E. T ´OTH AND T. WALDHAUSER we show that for systems of equations over a clone C , if all solution sets can bedescribed by closure under a clone D , then D must be the centralizer of C . Section 4contains the full description of finite lattices with Property (SDC): a finite lattice hasProperty (SDC) if and only if it is a Boolean lattice. In Section 5 finite semilatticeshaving Property (SDC) are described as semilattice reducts of distributive lattices.2. Preliminaries
Operations and clones.
Let A be an arbitrary set with at least two elements.By an operation on A we mean a map f : A n → A ; the positive integer n is called the arity of the operation f . The set of all operations on A is denoted by O A . For a set F ⊆ O A of operations, by F ( n ) we mean the set of n -ary members of F . In particular, O ( n ) A stands for the set of all n -ary operations on A .We will denote tuples by boldface letters, and we will use the corresponding plainletters with subscripts for the components of the tuples. For example, if a ∈ A n , then a i denotes the i -th component of a , i.e., a = ( a , . . . , a n ). In particular, if f ∈ O ( n ) A ,then f ( a ) is a short form for f ( a , . . . , a n ). If t (1) , . . . , t ( m ) ∈ A n and f ∈ O ( m ) A , then f ( t (1) , . . . , t ( m ) ) denotes the n -tuple obtained by applying f to the tuples t (1) , . . . , t ( m ) componentwise: f ( t (1) , . . . , t ( m ) ) = (cid:0) f ( t (1)1 , . . . , t ( m )1 ) , . . . , f ( t (1) n , . . . , t ( m ) n ) (cid:1) . We say that T ⊆ A n is closed under C , if for all m ∈ N , t (1) , . . . , t ( m ) ∈ T and for all f ∈ C ( m ) we have f ( t (1) , . . . , t ( m ) ) ∈ T .Let f ∈ O ( n ) A and g , . . . , g n ∈ O ( k ) A . By the composition of f by g , . . . , g n we meanthe operation h ∈ O ( k ) A defined by h ( x ) = f (cid:0) g ( x ) , . . . , g n ( x ) (cid:1) for all x ∈ A k . Now we present the precise definition of clones.
Definition 2.1. If C ⊆ O A is closed under composition and contains the projections e ( n ) i : ( x , . . . , x n ) (cid:55)→ x i for all 1 ≤ i ≤ n ∈ N , then C is said to be a clone (notation: C ≤ O A ).For an arbitrary set F of operations on A , there is a least clone [ F ] containing F ,called the clone generated by F . The elements of this clone are those operations thatcan be obtained from members of F and from projections by finitely many composi-tions. In other words, [ F ] is the set of term operations of the algebra A = ( A, F ).The set of all clones on A is a lattice under inclusion; the greatest element of thislattice is O A , and the least element is the trivial clone consisting of projections only.There are countably infinitely many clones on the two-element set; these have beendescribed by Post [11], hence the lattice of clones on { , } is called the Post lattice . If A is a finite set with at least three elements, then the clone lattice on A is of continuumcardinality [8], and it is a very difficult open problem to describe all clones on A evenfor | A | = 3.2.2. Centralizer clones.
We say that the operations f ∈ O ( n ) A and g ∈ O ( m ) A com-mute (notation: f ⊥ g ) if f (cid:0) g ( a , a , . . . , a m ) , . . . , g ( a n , a n , . . . , a nm ) (cid:1) = g (cid:0) f ( a , a , . . . , a n ) , . . . , f ( a m , a m , . . . , a nm ) (cid:1) holds for all a ij ∈ A (1 ≤ i ≤ n, ≤ j ≤ m ). This can be visualized as follows: forevery n × m matrix Q = ( a ij ), first applying g to the rows of Q and then applying f tothe resulting column vector yields the same result as first applying f to the columnsof Q and then applying g to the resulting row vector (see Figure 1). Definition 2.2.
For any F ⊆ O A , the set F ∗ := { g ∈ O A | f ⊥ g for all f ∈ F } iscalled the centralizer of F . OLUTION SETS OF SYSTEMS OF EQUATIONS OVER FINITE (SEMI)LATTICES 3 a . . . a m ... ... a n . . . a nm g −−−−→ (cid:121) f (cid:121) fg −−−−→ Figure 1.
Commutation of f and g .It is easy to verify that if f, g , . . . , g n all commute with an operation h , then thecomposition f ( g , . . . , g n ) also commutes with h . This implies that F ∗ is a clone forall F ⊆ O A (even if F itself is not a clone).Clones arising in this form are called primitive positive clones ; such clones seem tobe quite rare: there are only finitely many primitive positive clones over any finite set[2]. Example 2.3.
Let K be a field, and let L be the clone of all operations over K thatare represented by a linear polynomial: L := { a x + · · · + a k x k + c | k ≥ , a , . . . , a k , c ∈ K } . Since L is generated by the operations x + y , ax ( a ∈ K ) and the constants c ∈ K ,the centralizer L ∗ consists of those operations f over K that commute with x + y and ax (i.e., f is additive and homogeneous), and also commute with the constants (i.e., f ( c, . . . , c ) = c for all c ∈ K ): L ∗ := { a x + · · · + a k x k | k ≥ , a , . . . , a k ∈ K and a + · · · + a k = 1 } . Similarly, one can verify that L ∗ = L for the clone L := { a x + · · · + a k x k | k ≥ , a , . . . , a k ∈ K } . Equations and solution sets.
Let us fix a finite set A , a clone C ≤ O A and anatural number n . By an n -ary equation over C ( C -equation for short) we mean anequation of the form f ( x , . . . , x n ) = g ( x , . . . , x n ), where f, g ∈ C ( n ) . We will oftensimply write this equation as a pair ( f, g ). A system of C -equations is a finite set of C -equations of the same arity: E := (cid:8) ( f , g ) , . . . , ( f t , g t ) (cid:9) , where f i , g i ∈ C ( n ) ( i = 1 , . . . , t ) . Note that we consider only systems consisting of a finite number of equations. Thisdoes not restrict generality, since we are dealing only with finite algebras. We definethe set of solutions of E as the setSol( E ) := (cid:8) a ∈ A n | f i ( a ) = g i ( a ) for i = 1 , . . . , t (cid:9) . For a ∈ A n we denote by Eq C ( a ) the set of C -equations satisfied by a :Eq C ( a ) := (cid:8) ( f, g ) | f, g ∈ C ( n ) and f ( a ) = g ( a ) (cid:9) . Let T ⊆ A n be an arbitrary set of tuples. We denote by Eq C ( T ) the set of C -equationssatisfied by T : Eq C ( T ) := (cid:92) a ∈ T Eq C ( a ) . Remark 2.4.
For any given n ∈ N and C ≤ O A , the operators Sol and Eq C give riseto a Galois connection between sets of n -tuples and systems of n -ary equations. Inparticular, if T is the solution set of a system of equations (i.e., T is Galois closed),then T = Sol(Eq C ( T )); moreover, E = Eq C ( T ) is the largest system of equations with T = Sol( E ). E. T ´OTH AND T. WALDHAUSER
Example 2.5.
Considering the “linear” clones of Example 2.3, L -equations are linearequations and L -equations are homogeneous linear equations.In a previous paper [12] we proved that for any clone, the solution sets are closed un-der the centralizer of the clone. Furthermore, we proved the following theorem, whichcharacterizes solution sets of systems of equations over clones of Boolean functions. Theorem 2.6 ([12]) . For any clone of Boolean functions C ≤ O { , } and T ⊆ { , } n ,the following conditions are equivalent: (a) there is a system E of C -equations such that T = Sol( E ) ; (b) T is closed under C ∗ . Thus for two-element algebras, closure under the centralizer characterizes solutionsets. We will say that a clone C has Property (SDC), if this is true for the clone: Property (SDC).
The following are equivalent for all n ∈ N and T ⊆ A n :(a) there exists a system E of C -equations such that T = Sol( E );(b) the set T is closed under C ∗ .Here SDC is an abbreviation for “Solution sets are Definable by closure under theCentralizer”. In [12] we presented a clone on a three-element set that does not haveProperty (SDC), showing that in general this is not a trivial property.2.4. The Pol-Inv Galois connection.
For a positive integer h , a set ρ ⊆ A h iscalled an h -ary relation on A ; let R A denote the set of all relations on A . For any R ⊆ R A , let R ( h ) denote the h -ary part of R , i.e., the set of h -ary members of R .For a relation ρ ⊆ A h and operation f ∈ O ( n ) A , if for arbitrary tuples a (1) , . . . , a ( n ) ∈ ρ we have f ( a (1) , . . . , a ( n ) ) ∈ ρ , then we say that f is a polymorphism of ρ , or ρ isan invariant relation of f (or we also say that f preserves ρ ). We will denote this as f (cid:66) ρ . Note that f (cid:66) ρ is equivalent to ρ being closed under f (see Subsection 2.1).Preservation induces the so-called Pol - Inv
Galois connection . For any F ⊆ O A andfor any R ⊆ R A , let Inv ( F ) := { ρ ∈ R A | ∀ f ∈ F : f (cid:66) ρ } , andPol ( R ) := { f ∈ O A | ∀ ρ ∈ R : f (cid:66) ρ } . It is easy to verify that Pol ( R ) is a clone for all R ⊆ R A . Moreover, for every setof operations F on a finite set, the clone generated by F is [ F ] = Pol(Inv( F )) by theresults of Bodnarˇcuk, Kaluˇznin, Kotov, Romov and Geiger [1, 5].Given a set of relations R ⊆ R A , a primitive positive formula over R (pp. formulafor short) is a formula of the form(2.1) Φ( x , . . . , x n ) = ∃ y ∃ y . . . ∃ y m t ¯ j =1 ρ j (cid:0) z ( j )1 , . . . , z ( j ) r j (cid:1) , where ρ j ∈ R ( r j ) , and z ( j ) i ( j = 1 , . . . , t, and i = 1 , . . . , r j ) are variables from the set { x , . . . , x n , y , . . . , y m } . The setrel (Φ) := { ( a , . . . , a n ) | Φ( a , . . . , a n ) is true } is an n -ary relation, which is the relation defined by Φ. If R ⊆ R A , then let (cid:104) R (cid:105) ∃ denote the set of all relations that can be defined by a primitive positive formula over R ∪{ = } , and let (cid:104) R (cid:105) (cid:64) denote the set of all relations that can be defined by a quantifier-free primitive positive formula over R ∪ { = } . If R ⊆ R A contains the equality relationand R is closed under primitive positive definability, then we say that R is a relationalclone . The relational clone generated by R is (cid:104) R (cid:105) ∃ = Inv(Pol( R )) [1, 5].For f ∈ O ( n ) A , we define the following relation on A , called the graph of f : f • = { ( a , . . . , a n , b ) | f ( a , . . . , a n ) = b } ⊆ A n +1 . For F ⊆ O A , let F • = { f • | f ∈ F } . It is not hard to see that for any f ∈ O ( n ) A and g ∈ O ( m ) A the function f commutes with g if and only if f preserves the graph of g (or OLUTION SETS OF SYSTEMS OF EQUATIONS OVER FINITE (SEMI)LATTICES 5 equivalently, if and only if g preserves the graph of f ). Therefore for any F ⊆ O A wehave F ∗ = Pol( F • ). 3. Quantifier elimination
Let F ⊆ O A , then let F ◦ denote the set of all relations that are solution sets ofsome equation over F : F ◦ = (cid:8) Sol( f, g ) (cid:12)(cid:12) n ∈ N , f, g ∈ F ( n ) (cid:9) ⊆ R A . The following remark shows that the graph of an operation f ∈ F also belongs to F ◦ . Remark 3.1.
Let f ∈ O ( n ) A , and define (cid:101) f ∈ O ( n +1) A by (cid:101) f ( x , . . . , x n , x n +1 ) := f ( x , . . . , x n ). Then we haveSol (cid:0) (cid:101) f , e ( n +1) n +1 (cid:1) = (cid:8) ( a , . . . , a n , b ) ∈ A n +1 (cid:12)(cid:12) (cid:101) f ( a , . . . , a n , b ) = e ( n +1) n +1 ( a , . . . , a n , b ) (cid:9) = (cid:8) ( a , . . . , a n , b ) ∈ A n +1 (cid:12)(cid:12) f ( a , . . . , a n ) = b (cid:9) = f • . The following three lemmas prepare the proof of Theorem 3.6, which gives us anequivalent condition to Property (SDC) that we will use in sections 4 and 5.
Lemma 3.2.
For every clone C ≤ O A , we have C • ⊆ C ◦ and (cid:104) C • (cid:105) ∃ = (cid:104) C ◦ (cid:105) ∃ .Proof. In accordance with Remark 3.1, for all f ∈ C we have Sol( (cid:101) f , e ( n +1) n +1 ) = f • ∈ C ◦ . Therefore C • ⊆ C ◦ , which implies that (cid:104) C • (cid:105) ∃ ⊆ (cid:104) C ◦ (cid:105) ∃ . To prove the reversedcontainment, let us consider an arbitrary relation ρ = Sol( f, g ) ∈ C ◦ with f, g ∈ C ( n ) .Then, for any ( x , . . . , x n ) ∈ A n , we have( x , . . . , x n ) ∈ ρ ⇐⇒ f ( x , . . . , x n ) = g ( x , . . . , x n ) ⇐⇒ ∃ y : f ( x , . . . , x n ) = y & g ( x , . . . , x n ) = y ⇐⇒ ∃ y : ( x , . . . , x n , y ) ∈ f • & ( x , . . . , x n , y ) ∈ g • . This means that ρ can be defined by a pp. formula over { f • , g • } , therefore ρ ∈ (cid:104) C • (cid:105) ∃ .Thus, we obtain C ◦ ⊆ (cid:104) C • (cid:105) ∃ , and this implies that (cid:104) C ◦ (cid:105) ∃ ⊆ (cid:104)(cid:104) C • (cid:105) ∃ (cid:105) ∃ = (cid:104) C • (cid:105) ∃ .Therefore (cid:104) C • (cid:105) ∃ = (cid:104) C ◦ (cid:105) ∃ . (cid:3) Lemma 3.3.
For every clone C ≤ O A and T ⊆ A n , there is a system E of C -equationssuch that T = Sol( E ) if and only if T ∈ (cid:104) C ◦ (cid:105) (cid:64) .Proof. Let Φ be an arbitrary quantifier-free pp. formula over C ◦ . By definition, Φ isof the formΦ( x , . . . , x n ) = t ¯ j =1 Sol( f j , g j ) = t ¯ j =1 (cid:16) f j (cid:0) z ( j )1 , . . . , z ( j ) r j (cid:1) = g j (cid:0) z ( j )1 , . . . , z ( j ) r j (cid:1)(cid:17) , where n, t ∈ N , f j , g j ∈ C ( r j ) and z ( j )1 , . . . , z ( j ) r j ∈ { x , . . . , x n } for all j = 1 , . . . , t .We define the operations (cid:101) f j ( x , . . . , x n ) := f j ( z ( j )1 , . . . , z ( j ) r j ) and (cid:101) g j ( x , . . . , x n ) := g j ( z ( j )1 , . . . , z ( j ) r j ) (by identifying variables and by adding fictitious variables) for all j = 1 , . . . , t . Then Φ is equivalent to the formulaΨ( x , . . . , x n ) = t ¯ j =1 (cid:16) (cid:101) f j (cid:0) x , . . . , x n (cid:1) = (cid:101) g j (cid:0) x , . . . , x n (cid:1)(cid:17) , and (cid:101) f j , (cid:101) g j ∈ C ( n ) for all j = 1 , . . . , t . Since Φ and Ψ are equivalent, they define thesame set T ⊆ A n , and it is obvious that the set defined by Ψ is the solution set ofthe system { ( (cid:101) f , (cid:101) g ) , . . . , ( (cid:101) f t , (cid:101) g t ) } . Conversely, it is clear that every solution set can bedefined by a quantifier-free pp. formula of the form of Ψ. (cid:3) Lemma 3.4.
For every clone C ≤ O A , we have Inv ( C ∗ ) = (cid:104) C • (cid:105) ∃ . Consequently, aset T ⊆ A n is closed under C ∗ if and only if T ∈ (cid:104) C ◦ (cid:105) ∃ . E. T ´OTH AND T. WALDHAUSER
Proof.
From Section 2 using that F ∗ = Pol( F • ) and that Inv(Pol( R )) = (cid:104) R (cid:105) ∃ , wehave Inv ( C ∗ ) = Inv (Pol ( C • )) = (cid:104) C • (cid:105) ∃ . The second statement of the lemma follows immediately from Lemma 3.2 by observingthat T is closed under C ∗ if and only if T ∈ Inv( C ∗ ). (cid:3) Theorem 3.5 ([12]) . For every clone C ≤ O A and T ⊆ A n , if there is a system E of C -equations such that T = Sol( E ) , then T is closed under C ∗ .Proof. Let C ≤ O A , T ⊆ A n , and let E be a system of C -equations and T = Sol( E ).By Lemma 3.3 we have T ∈ (cid:104) C ◦ (cid:105) (cid:64) ⊆ (cid:104) C ◦ (cid:105) ∃ . Using Lemma 3.4, this means that T isclosed under C ∗ . (cid:3) The previous theorem shows that in Property (SDC), condition (a) implies (b).Therefore, for all clones C ≤ O A , it suffices to investigate the implication (b) = ⇒ (a).As a consequence of lemmas 3.2, 3.3 and 3.4, we obtain the promised equivalentreformulation of Property (SDC) in terms of quantifier elimination. Theorem 3.6.
For every clone C ≤ O A , the following five conditions are equivalent: (i) C has Property (SDC) ; (ii) (cid:104) C ◦ (cid:105) (cid:64) = Inv ( C ∗ ) ; (iii) (cid:104) C ◦ (cid:105) (cid:64) = (cid:104) C ◦ (cid:105) ∃ ; (iv) every primitive positive formula over C ◦ is equivalent to a quantifier-free prim-itive positive formula over C ◦ ; (v) (cid:104) C ◦ (cid:105) (cid:64) is a relational clone.Proof. (i) ⇐⇒ (ii): By Lemma 3.3, T is the solution set of some system of equationsover C if and only if T ∈ (cid:104) C ◦ (cid:105) (cid:64) .(ii) ⇐⇒ (iii): This follows from (the proof of) Lemma 3.4.(iii) ⇐⇒ (iv): This is trivial.(iii) ⇐⇒ (v): This follows from the fact that the relational clone generated by (cid:104) C ◦ (cid:105) (cid:64) is (cid:104) C ◦ (cid:105) ∃ . (cid:3) In the following corollary we will see that Theorem 3.6 implies that C ∗ is the onlyclone that can describe solution sets over C (if there is such a clone at all). Thus, theabbreviation SDC can also stand for “Solution sets are Definable by closure under anyClone”. Corollary 3.7.
Let C ≤ O A be a clone, and assume that there is a clone D such thatfor all n ∈ N and T ⊆ A n the following equivalence holds: T is the solution set of a system of C -equations ⇐⇒ T is closed under D. Then we have D = C ∗ .Proof. The condition in the corollary gives us by Lemma 3.3 that for all T ⊆ A n ,we have T ∈ (cid:104) C ◦ (cid:105) (cid:64) if and only if T ∈ Inv ( D ). This means that (cid:104) C ◦ (cid:105) (cid:64) = Inv ( D ),thus (cid:104) C ◦ (cid:105) (cid:64) is a relational clone. Therefore, by Theorem 3.6 this is equivalent to thecondition Inv ( C ∗ ) = (cid:104) C ◦ (cid:105) (cid:64) = Inv ( D ). Applying the operator Pol to the last equalitywe get that C ∗ = Pol (Inv ( C ∗ )) = Pol (Inv ( D )) = D. (cid:3) Systems of equations over lattices
In this, and in the following section L = ( L, ∧ , ∨ ) denotes a finite lattice, with meetoperation ∧ and join operation ∨ . Furthermore, 0 L denotes the least and 1 L denotesthe greatest element of L (that is, 0 L = (cid:86) L and 1 L = (cid:87) L ).The following lemma shows that Property (SDC) does not hold for non-distributivelattices, i.e., solution sets of systems of equations over a non-distributive lattice cannot be characterized via closure conditions. OLUTION SETS OF SYSTEMS OF EQUATIONS OVER FINITE (SEMI)LATTICES 7 u b a a ub = x y x ∧ y x ∨ yx − ( a, b ) ( b, a ) ( a, b ) y − − ( a, b ) ( b, a ) x ∧ y − − − ( a, b ) x ∨ y − − − − Figure 2.
Counterexamples showing that these equations do notbelong to E . Lemma 4.1.
Let L = ( L, ∧ , ∨ ) be a finite lattice. If Property (SDC) holds for C =[ ∧ , ∨ ] , then L is a distributive lattice.Proof. Let L = ( L, ∧ , ∨ ) be a non-distributive finite lattice and C = [ ∧ , ∨ ] ≤ O L . ByLemma 3.4, the set T = { ( x, y ) | ∃ u ∈ L : u ∧ x = u ∧ y and u ∨ x = u ∨ y } ⊆ L is closed under C ∗ . We prove that T is not the solution set of a system of equations over C , hence Property (SDC) does not hold for C . Suppose that there exists a systemof C -equations E such that T = Sol( E ). Since L is not distributive, by Birkhoff’stheorem we know that there is a sublattice of L , which is isomorphic either to N or M . Now neither of the equations x = y ( ⇐⇒ x ∧ y = x ∨ y ) , x = x ∧ y, x = x ∨ y, y = x ∧ y, y = x ∨ y belong to E ; we prove this by presenting a counterexample for each equation. Thesecounterexamples are shown in Figure 2, where we choose the elements a and b aspresented in the figure. (Note that an element u , chosen like on the figure, showsthat ( a, b ) , ( b, a ) ∈ T . In the table, the entry ( x , y ) in the line starting with theterm s ( x, y ) and column starting with the term t ( x, y ) witnesses that ( x , y ) is not asolution of s ( x, y ) = t ( x, y ).) There are no other non-trivial 2-variable equations over C , therefore we get that T satifies only trivial equations, hence T = L . This is acontradiction, since (0 L , L ) / ∈ T . (cid:3) The following lemma will help us prove that Property (SDC) can only hold forBoolean lattices. Before the lemma, for a distributive lattice L we define the median of the elements x, y, z ∈ L as m ( x, y, z ) = ( x ∧ y ) ∨ ( x ∧ z ) ∨ ( y ∧ z ) = ( x ∨ y ) ∧ ( x ∨ z ) ∧ ( y ∨ z ) . Lemma 4.2.
Let L = ( L, ∧ , ∨ ) be a distributive lattice, and for all x, y, z, u ∈ L let p ( x, y, z, u ) = ( x ∧ y ) ∨ ( x ∧ z ) ∨ ( y ∧ z ) ∨ ( u ∧ x ) ∨ ( u ∧ y ) ∨ ( u ∧ z ) . Then for all x, y, z, u ∈ L we have p ( x, y, z, u ) = x ∨ y ∨ z ∨ u ⇐⇒ m ( x, y, z ) ∨ u = x ∨ y ∨ z. Proof.
Let x, y, z, u ∈ L be arbitrary elements. Let us denote m ( x, y, z ) simply by m and p ( x, y, z, u ) by p for better readability.First let us suppose that p = x ∨ y ∨ z ∨ u . It is easy to see that p ≤ x ∨ y ∨ z alwaysholds (since every meet in p is less than or equal to x ∨ y ∨ z ). Since p = x ∨ y ∨ z ∨ u ,we get that p ≤ x ∨ y ∨ z ≤ x ∨ y ∨ z ∨ u = p , hence p = x ∨ y ∨ z . Observe thatby the distributivity of L , p can be rewritten as p = m ∨ ( u ∧ ( x ∨ y ∨ z )), and fromthe previous chain of inequalities we can see that u ≤ x ∨ y ∨ z , therefore we have p = m ∨ u . Thus m ∨ u = p = x ∨ y ∨ z .For the other direction suppose that m ∨ u = x ∨ y ∨ z . Using that L is distributive,we get that p = m ∨ ( u ∧ ( x ∨ y ∨ z )) = ( m ∨ u ) ∧ ( m ∨ ( x ∨ y ∨ z )), and by the assumptionthis implies that p = x ∨ y ∨ z . Our assumption also implies that u ≤ x ∨ y ∨ z , thereforewe have p = x ∨ y ∨ z ∨ u . (cid:3) E. T ´OTH AND T. WALDHAUSER
Theorem 4.3.
Let L = ( L, ∧ , ∨ ) be a finite distributive lattice. If Property (SDC) holds for C = [ ∧ , ∨ ] , then L is a Boolean lattice.Proof. Let L = ( L, ∧ , ∨ ) be a finite distributive lattice and let C = [ ∧ , ∨ ] ≤ O L .Since L is distributive, by Birkhoff’s representation theorem L can be embedded intoa Boolean lattice B , hence we may suppose without loss of generality that L is alreadya sublattice of B . We can also assume that 0 L = 0 B and 1 L = 1 B . Let us denote thecomplement of an element x ∈ B by x (cid:48) . We define the dual of p = p ( x, y, z, u ) (fromLemma 4.2) as p d = q = q ( x, y, z, u ), i.e., q ( x, y, z, u ) = ( x ∨ y ) ∧ ( x ∨ z ) ∧ ( y ∨ z ) ∧ ( u ∨ x ) ∧ ( u ∨ y ) ∧ ( u ∨ z ) . Let T be the following set: T = (cid:8) ( x, y, z ) ∈ L (cid:12)(cid:12) ∃ u ∈ L : p ( x, y, z, u ) = x ∨ y ∨ z ∨ u and q ( x, y, z, u ) = x ∧ y ∧ z ∧ u (cid:9) . By Lemma 3.4, the set T is closed under C ∗ . Let ( x, y, z ) ∈ T be arbitrary withan element u ∈ L witnessing that ( x, y, z ) ∈ T . From Lemma 4.2 it follows that p ( x, y, z, u ) = x ∨ y ∨ z ∨ u if and only if m ∨ u = x ∨ y ∨ z . Meeting both sides of thelatter equality by m (cid:48) , we get(4.1) u ∧ m (cid:48) = ( m ∧ m (cid:48) ) ∨ ( u ∧ m (cid:48) ) = ( m ∨ u ) ∧ m (cid:48) = ( x ∨ y ∨ z ) ∧ m (cid:48) . By the dual of Lemma 4.2, we know that q ( x, y, z, u ) = x ∧ y ∧ z ∧ u if and only if m ∧ u = x ∧ y ∧ z . Then joining the last equality and (4.1), we get that u = u ∧ L = u ∧ ( m (cid:48) ∨ m ) = ( u ∧ m (cid:48) ) ∨ ( u ∧ m )= (( x ∨ y ∨ z ) ∧ m (cid:48) ) ∨ ( x ∧ y ∧ z ) . It is not hard to derive from the defining identities of Boolean algebras that the latterformula is in fact the symmetric difference x (cid:77) y (cid:77) z in B . Alternatively, usingStone’s representation theorem for Boolean algebras, we may assume that x , y and z are sets, and that the operations ∧ , ∨ , (cid:48) are the set-theoretic intersection, union andcomplementation. Then m corresponds to the set of elements that belong to at leasttwo of the sets x , y and z . Thus ( x ∨ y ∨ z ) ∧ m (cid:48) consists of those elements that belongto exactly one of x , y and z , and (( x ∨ y ∨ z ) ∧ m (cid:48) ) ∨ ( x ∧ y ∧ z ) contains those elementsthat belong to one or three of the sets x , y and z , and this is indeed x (cid:77) y (cid:77) z in B .We have proved that the element u witnessing that ( x, y, z ) ∈ T can only be x (cid:77) y (cid:77) z :(4.2) ∀ x, y, z ∈ L : ( x, y, z ) ∈ T ⇐⇒ ∃ u ∈ L : u = x (cid:77) y (cid:77) z ⇐⇒ x (cid:77) y (cid:77) z ∈ L. It is easy to see that { L , L } ⊆ T , and using the main theorem of [6], we get thatif ( f, g ) ∈ Eq( T ), then f = g must hold. (In our case this theorem says that everyterm function of L is uniquely determined by its restriction to { , } .) Therefore onlytrivial equations can appear in Eq( T ), hence T = L . Then (4.2) implies that L isclosed under the ternary operation x (cid:77) y (cid:77) z . In particular, for any x ∈ L we have x (cid:77) (cid:77) x (cid:48) ∈ L , which means that L is a Boolean lattice. (cid:3) We will need the following lemmas for the proof of Theorem 4.7, which states thatBoolean lattices have Property (SDC). This will complete the determination of latticeswith Property (SDC).
Lemma 4.4.
Let L = ( L, ∧ , ∨ ) be a finite distributive lattice and let C = [ ∧ , ∨ ] ≤ O L .Then every system of C -equations is equivalent to a system of inequalities { p ≤ q , . . . , p l ≤ q l } , where p i ∈ [ ∧ ] and q i ∈ [ ∨ ] ( i = 1 , . . . , l ).Proof. Let L = ( L, ∧ , ∨ ) be a finite distributive lattice, let C = [ ∧ , ∨ ] ≤ O L and let E = { f = g , . . . , f t = g t } be a system of C -equations. For arbitrary a, b ∈ L we have a = b if and only if a ≤ b and b ≤ a , therefore E is equivalent to the system of inequalities E (cid:48) = { f ≤ g , g ≤ f , . . . , f t ≤ g t , g t ≤ f t } . OLUTION SETS OF SYSTEMS OF EQUATIONS OVER FINITE (SEMI)LATTICES 9
Denote the disjunctive normal forms of the left hand sides of the inequalities in E (cid:48) as DN F j , and denote the conjunctive normal forms of the right hand sides of theinequalities in E (cid:48) as CN F j ( j = 1 , . . . , t ). Then E (cid:48) is equivalent to the system ofinequalities { DN F ≤ CN F , . . . , DN F t ≤ CN F t } . Each
DN F j is a join of some meets, and each CN F j is a meet of some joins. Therefore,for every j the inequality DN F j ≤ CN F j holds if and only if every meet in DN F j is less than or equal to every join in CN F j . This means that there exists a systemof inequalities { p ≤ q , . . . , p l ≤ q l } equivalent to E , such that p i ∈ [ ∧ ] and q i ∈ [ ∨ ]( i = 1 , . . . , l ). (cid:3) Lemma 4.5.
Let B = ( B, ∧ , ∨ , (cid:48) ) be a Boolean algebra. Then for every a, b, c, d, u ∈ B ,we have (i) a ∧ u ≤ b ⇐⇒ u ≤ a (cid:48) ∨ b ; (ii) b ≤ a ∨ u ⇐⇒ u ≥ a (cid:48) ∧ b ; (iii) a ∧ b (cid:48) ≤ c (cid:48) ∨ d ⇐⇒ a ∧ c ≤ b ∨ d .Proof. Let a, b, c, d, u ∈ B be arbitrary elements. For the proof of (i) let us firstsuppose that a ∧ u ≤ b . Joining both sides of the inequality by a (cid:48) , we get a (cid:48) ∨ ( a ∧ u ) = ( a (cid:48) ∨ a ) ∧ ( a (cid:48) ∨ u ) = 1 B ∧ ( a (cid:48) ∨ u ) = a (cid:48) ∨ u ≤ a (cid:48) ∨ b, and from this, u ≤ a (cid:48) ∨ b follows. For the other direction, if u ≤ a (cid:48) ∨ b holds, thenmeeting both sides by a , we get that a ∧ u ≤ a ∧ ( a (cid:48) ∨ b ) = ( a ∧ a (cid:48) ) ∨ ( a ∧ b ) = 0 B ∨ ( a ∧ b ) = a ∧ b, and from this, a ∧ u ≤ b follows.The second statement is the dual of (i).For the proof of (iii) let us use (i) with u = a ∧ b (cid:48) , and then we get that a ∧ b (cid:48) ≤ c (cid:48) ∨ d ⇐⇒ c ∧ ( a ∧ b (cid:48) ) = ( c ∧ a ) ∧ b (cid:48) ≤ d. Then using (ii) with u = d , we get( c ∧ a ) ∧ b (cid:48) ≤ d ⇐⇒ c ∧ a ≤ b ∨ d, which proves (iii). (cid:3) Helly’s theorem from convex geometry states that if we have k ( > d ) convex sets in R d , such that any d + 1 of them have a nonempty intersection, then the intersectionof all k sets is nonempty as well. The following lemma says something similar forintervals in lattices (with d = 1). Lemma 4.6.
Let L = ( L, ∧ , ∨ ) be a lattice, c i , d i ∈ L ( i = 1 , . . . , k ). Then we have k (cid:92) i =1 [ c i , d i ] (cid:54) = ∅ ⇐⇒ ∀ i, j ∈ { , . . . , k } : c i ≤ d j . Proof.
Let L = ( L, ∧ , ∨ ) be a lattice, and c i , d i ∈ L ( i = 1 , . . . , k ). Then obviously, k (cid:92) i =1 [ c i , d i ] = [ c ∨ · · · ∨ c k , d ∧ · · · ∧ d k ] , which is nonempty if and only if c ∨ · · · ∨ c k ≤ d ∧ · · · ∧ d k , which holds if and onlyif c i ≤ d j for all i, j ∈ { , ..., k } . (cid:3) The last step in the characterization of finite lattices having Property (SDC) is toshow that Boolean lattices do indeed have Property (SDC). For proving this, we willuse the equivalence of this property with the quantifier-eliminability for pp. formulasover C ◦ = [ ∧ , ∨ ] ◦ (see Theorem 3.6). Theorem 4.7. If L = ( L, ∧ , ∨ ) is a finite Boolean lattice, then Property (SDC) holdsfor C = [ ∧ , ∨ ] . Proof.
Let L = ( L, ∧ , ∨ ) be a finite Boolean lattice, and let C = [ ∧ , ∨ ]. Let us denotethe complement of an element x ∈ L by x (cid:48) . By Theorem 3.6, Property (SDC) holdsfor C if and only if any pp. formula over C ◦ is equivalent to a quantifier-free pp.formula. Let us consider a pp. formula with a single quantifier:(4.3) Φ( x , . . . , x n ) = ∃ u t ¯ j =1 ρ j (cid:0) z ( j )1 , . . . , z ( j ) r j (cid:1) , where ρ j ∈ ( C ◦ ) ( r j ) , and z ( j ) i ( j = 1 , . . . , t, and i = 1 , . . . , r j ) are variables fromthe set { x , . . . , x n , u } . We will show that Φ is equivalent to a quantifier-free pp.formula, and thus (by iterating this argument) every pp. formula is equivalent to aquantifier-free pp. formula. By Lemma 4.4, we can rewrite Φ to an equivalent formula ∃ u l ¯ i =1 ( p i ≤ q i ) , where p i ∈ [ ∧ ] and q i ∈ [ ∨ ] ( i = 1 , . . . , l ). Let a i denote the meet of all variablesfrom { x , . . . , x n } appearing in p i , and let b i denote the join of all variables from { x , . . . , x n } appearing in q i . Then we can distinguish four cases for the i -th inequality:(0) If u does not appear in the inequality, then the inequality is of the form a i ≤ b i .(1) If u appears only on the left hand side of the inequality, then the inequalityis of the form a i ∧ u ≤ b i .(2) If u appears only on the right hand side of the inequality, then the inequalityis of the form a i ≤ b i ∨ u .(3) If u appears on both sides of the inequality, then the inequality is of the form a i ∧ u ≤ b i ∨ u , which always holds, since a i ∧ u ≤ u ≤ b i ∨ u .Let I j denote the following set of indices: I j = { i | the inequality p i ≤ q i belongs to case (j) } for j = 0 , , ,
3. The only cases we have to investigate are case (1) and case (2) (since u does not appear in case (0) and in case (3) there are only trivial inequalities). ByLemma 4.5,for i ∈ I we have a i ∧ u ≤ b i ⇐⇒ u ≤ a (cid:48) i ∨ b i ⇐⇒ u ∈ [0 L , a (cid:48) i ∨ b i ] =: [ c i , d i ];for i ∈ I we have a i ≤ b i ∨ u ⇐⇒ u ≥ b (cid:48) i ∧ a i ⇐⇒ u ∈ [ a i ∧ b (cid:48) i , L ] =: [ c i , d i ] . Then we have ∃ u ∀ i ∈ I ∪ I : p i ≤ q i ⇐⇒ (cid:92) i ∈ I ∪ I [ c i , d i ] (cid:54) = ∅ ⇐⇒ ∀ i, j ∈ I ∪ I : c i ≤ d j by Lemma 4.6. Since u does not appear in the condition above, in principle, thequantifier has been eliminated. However, our formula still involves complements.Therefore, we use Lemma 4.5 to rewrite the formula. The only non-trivial case is if c i (cid:54) = 0 L and d j (cid:54) = 1 L , that is, c i = a i ∧ b (cid:48) i and d j = a (cid:48) j ∨ b j ( i ∈ I , j ∈ I ). In this case c i ≤ d j if and only if a i ∧ a j ≤ b i ∨ b j by Lemma 4.5.Summarizing the observations above, we haveΦ( x , . . . , x n ) ≡ ∃ u l ¯ i =1 ( p i ≤ q i ) ≡ ¯ i ∈ I ( a i ≤ b i ) ¯ i,j ∈ I ∪ I ( c i ≤ d j ) ≡ ¯ i ∈ I ( a i ≤ b i ) ¯ i ∈ I ,j ∈ I ( a i ∧ a j ≤ b i ∨ b j ) , which is equivalent to a quantifier-free pp. formula over [ ∧ , ∨ ] (since for all x, y ∈ L ,we have x ≤ y if and only if x = x ∧ y ). (cid:3) We can summarize the results of this section in the following theorem, which is acorollary of Lemma 4.1, Theorem 4.3 and Theorem 4.7.
Theorem 4.8.
A finite lattice has Property (SDC) if and only if it is a Boolean lattice.
OLUTION SETS OF SYSTEMS OF EQUATIONS OVER FINITE (SEMI)LATTICES 11 = x y x ∧ yx − ( a, M ) ( a, M ) y − − (0 M , a ) x ∧ y − − − Table 1.
Counterexamples showing that these equations do not be-long to E .This means that for any finite lattice L = ( L, ∧ , ∨ ), solution sets of systems ofequations over L can be characterized (via closure conditions) if and only if L is aBoolean lattice. 5. Systems of equations over semilattices
Similarly to Section 4, in this section M = ( M, ∧ ) denotes a finite semilattice withmeet operation ∧ and least element 0 M . Lemma 5.1.
Let M = ( M, ∧ ) be a finite semilattice. If M has no greatest element,then Property (SDC) does not hold for C = [ ∧ ] .Proof. Let M = ( M, ∧ ) be a finite semilattice with no greatest element, and let C =[ ∧ ] ≤ O M . The set T = { ( x, y ) | ∃ u ∈ L : x ∧ u = x and y ∧ u = y } == { ( x, y ) | ∃ u ∈ L : x ≤ u and y ≤ u } ⊆ M is closed under C ∗ by Lemma 3.4. Similarly to Lemma 4.1, we will prove that T is notthe solution set of any system of equations over C . Suppose that there exists a systemof C -equations E such that T = Sol( E ). There are only three nontrivial 2-variableequations over C : x = y, x ∧ y = x, x ∧ y = y. As in Lemma 4.1, we prove that none of these equations can appear in E by presentingcounterexamples to them (see Table 1). Note that since M is finite and it has nogreatest element, there exist maximal elements a (cid:54) = b in M . Then we have that onlytrivial equations can appear in E , thus T = M . But this is a contradiction, since( a, b ) / ∈ T . (cid:3) If a finite semilattice M = ( M, ∧ ) has a greatest element, then for all ( a, b ) ∈ M ,the set H = { x ∈ M | a ≤ x and b ≤ x } is not empty. Since M is a finite semilattice,it follows that (cid:86) H exists for all ( a, b ) ∈ M . This means that we can define a joinoperation ∨ on M , such that L = ( L, ∧ , ∨ ) is a lattice (with L = M ). Therefore, fromnow on it suffices to investigate lattices (but the clone we use for the equations is still C = [ ∧ ]).The following theorem shows that Property (SDC) does not hold for non-distributivelattices (regarded as semilattices), i.e., solution sets of systems of equations over a non-distributive lattice (as a semilattice) can not be characterized via closure conditions. Remark 5.2.
A meet semilattice M is distributive if for any a, b , b ∈ M , the in-equality a ≥ b ∧ b implies that there exist a , a ∈ M such that a ≥ b , a ≥ b and a = a ∧ a (see Section 5.1 in Chapter II of [7]). From Lemma 184 of [7] itfollows that a finite semilattice is distributive if and only if it is a semilattice reductof a distributive lattice. Theorem 5.3.
Let L = ( L, ∧ , ∨ ) be a finite lattice. If L is not distributive, thenProperty (SDC) does not hold for C = [ ∧ ] .Proof. Let L = ( L, ∧ , ∨ ) be a finite lattice and let C = [ ∧ ] ≤ O L . Since L is notdistributive, we know that there exists a sublattice of L isomorphic to either N or M . Let us denote these two cases as ( N ) and ( M ), respectively. The figures andtables we use in this proof can be found in the Appendix. Let T be the set T = { ( x, y, z ) ∈ L | ∃ u ∈ L : x ∧ y = u ∧ y and u ∧ x = x and u ∧ z = z } = { ( x, y, z ) ∈ L | ∃ u ∈ L : x ∧ y = u ∧ y and u ≥ x and u ≥ z } , which is closed under C ∗ by Lemma 3.4. As in Lemma 4.1, we will prove that T isnot the solution set of any system of equations over C . Similarly to Lemma 4.1, wepresent counterexamples to nontrivial equations, the only difference is that here weprove that there can be only one nontrivial equation satisfied by T (see tables 2 and 3for case ( N ) and ( M ), respectively). We choose the elements a, b and c as presentedin Figure 3 for case ( N ), and in Figure 4 for case ( M ). (Note that an element u ,chosen like on the figures, shows that in case ( N ) we have ( a, c, b ) , ( b, a, c ) ∈ T , andin case ( M ) we have ( a, b, c ) , ( a, c, b ) ∈ T .)So now we have that in both cases the only nontrivial equation that T can satisfyis the equation y ∧ z = x ∧ y ∧ z . One can verify that this equation holds on T : if( x, y, z ) ∈ T , then we have x ∧ y = u ∧ y ≥ z ∧ y = ⇒ x ∧ y ∧ z ≥ y ∧ z, which implies that y ∧ z = x ∧ y ∧ z . Therefore, we can conclude that the only nontrivialequation in Eq( T ) is y ∧ z = x ∧ y ∧ z . We will prove that T is not the solution setof any system of equations by presenting a tuple ( x , y , z ) ∈ Sol(Eq( T )) \ T (cf.Remark 2.4). Since there exists a sublattice of L isomorphic to N or M , there existsa tuple ( x , y , z ) as shown in Figure 5, which satisfies y ∧ z = x ∧ y ∧ z , thus( x , y , z ) ∈ Sol(Eq( T )). However, one can easily verify that ( x , y , z ) does notbelong to T . Indeed, suppose that ( x , y , z ) ∈ T , then there exists u ∈ L such that u ≥ x , u ≥ z and x ∧ y = u ∧ y . But then we have u ≥ x ∨ z > y (since N or M is a sublattice), therefore x ∧ y < u ∧ y = y gives us a contradiction. Thus, T (cid:54) = Sol(Eq( T )), hence, by Remark 2.4, T is not the solution set of any system ofequations over C . (cid:3) Lemma 5.1 and Theorem 5.3 prove that if M = ( M, ∧ ) has Property (SDC), thenit is the semilattice reduct of a distributive lattice L = ( L, ∧ , ∨ ). To complete thecharacterization of finite semilattices with Property (SDC), we prove that the clone[ ∧ ] has Property (SDC) whenever ∧ is the meet operation of a finite distributivelattice. Theorem 5.4. If L = ( L, ∧ , ∨ ) is a finite distributive lattice, then Property (SDC) holds for C = [ ∧ ] .Proof. Let L = ( L, ∧ , ∨ ) be a finite distributive lattice and C = [ ∧ ] ≤ O L . Since L is distributive, by Birkhoff’s representation theorem L can be embedded into aBoolean lattice B , hence we may suppose without loss of generality that L is alreadya sublattice of B . We can also assume that 0 L = 0 B and 1 L = 1 B . Let us denote thecomplement of an element x ∈ B by x (cid:48) . By Theorem 3.6, Property (SDC) holds for C if and only if any pp. formula over C ◦ is equivalent to a quantifier-free pp. formula.Similarly to the proof of Theorem 4.7, it suffices to consider pp. formulas with a singleexistential quantifier. Let(5.1) Φ( x , . . . , x n ) = ∃ u t ¯ j =1 ρ j (cid:0) z ( j )1 , . . . , z ( j ) r j (cid:1) , where ρ j ∈ ( C ◦ ) ( r j ) , and z ( j ) i ( j = 1 , . . . , t, and i = 1 , . . . , r j ) are variables from theset { x , . . . , x n , u } . We will show that Φ is equivalent to a quantifier-free pp. formula.Since for all a, b ∈ L we have a = b if and only if a ≤ b and b ≤ a , we can rewriteΦ to an equivalent formula ∃ u l ¯ i =1 ( p i ≤ q i ) , OLUTION SETS OF SYSTEMS OF EQUATIONS OVER FINITE (SEMI)LATTICES 13 where p i , q i ∈ [ ∧ ] ( i = 1 , . . . , l ). Let a i denote the meet of all variables from { x , . . . , x n } appearing in p i , and let b i denote the meet of all variables from { x , . . . , x n } appearingin q i . Then we can distinguish four cases for the i -th inequality:(0) If u does not appear in the inequality, then the inequality is of the form a i ≤ b i .(1) If u appears only on the left hand side of the inequality, then the inequalityis of the form a i ∧ u ≤ b i .(2) If u appears only on the right hand side of the inequality, then the inequalityis of the form a i ≤ b i ∧ u , which holds if and only if a i ≤ b i and a i ≤ u .(3) If u appears on both sides of the inequality, then the inequality is of the form a i ∧ u ≤ b i ∧ u , which holds if and only if a i ∧ u ≤ b i and a i ∧ u ≤ u , that is, a i ∧ u ≤ b i .Let I j denote the following set of indices: I j = { i | the inequality p i ≤ q i belongs to case (j) } for j = 0 , , ,
3. We investigate only cases (1), (2) and (3), since u does not appear incase (0). Moreover; in case (2), we only have to deal with the inequality a i ≤ u , since u does not appear in the inequality a i ≤ b i . By Lemma 4.5,for i ∈ I we have a i ∧ u ≤ b i ⇐⇒ u ≤ a (cid:48) i ∨ b i ⇐⇒ u ∈ [0 L , a (cid:48) i ∨ b i ] =: [ c i , d i ];for i ∈ I we have a i ≤ u ⇐⇒ u ∈ [ a i , L ] =: [ c i , d i ];for i ∈ I we have a i ∧ u ≤ b i ⇐⇒ u ≤ a (cid:48) i ∨ b i ⇐⇒ u ∈ [0 L , a (cid:48) i ∨ b i ] =: [ c i , d i ] . Then we have (cid:92) i ∈ I ∪ I ∪ I [ c i , d i ] (cid:54) = ∅ ⇐⇒ ∀ i, j ∈ I ∪ I ∪ I : c i ≤ d j by Lemma 4.6. Just as in the proof of Theorem 4.7, we apply Lemma 4.5 to eliminatecomplements and joins from the formula above. The only interesting case is if c i (cid:54) = 0 L and d j (cid:54) = 1 L , that is, c i = a i and d j = a (cid:48) j ∨ b j ( i ∈ I , j ∈ I ∪ I ). In this case c i ≤ d j if and only if a i ≤ a (cid:48) j ∨ b j , which holds if and only if a i ∧ a j ≤ b j by Lemma 4.5 (with u = a i ).Summarizing the observations above, we haveΦ( x , . . . , x n ) ≡ ∃ u l ¯ i =1 ( p i ≤ q i ) ≡ ¯ i ∈ I ∪ I ( a i ≤ b i ) ¯ i,j ∈ I ∪ I ∪ I ( c i ≤ d j ) ≡ ¯ i ∈ I ∪ I ( a i ≤ b i ) ¯ i ∈ I ,j ∈ I ∪ I ( a i ∧ a j ≤ b j ) , which is equivalent to a quantifier-free pp. formula over [ ∧ ] (since for all x, y ∈ L , wehave x ≤ y if and only if x = x ∧ y ). (cid:3) We can summarize the results of this section in the following theorem, which is acorollary of Lemma 5.1, and theorems 5.3 and 5.4.
Theorem 5.5.
A finite semilattice has Property (SDC) if and only if it is distributive.
This means that for any finite semilattice M , solution sets of systems of equationsover M can be characterized (via closure conditions) if and only if M is a semilatticereduct of a distributive lattice (see Remark 5.2).6. Concluding remarks
We have characterized finite lattices and semilattices having Property (SDC). Asa natural continuation of these investigations, one could aim at describing all finitealgebras (clones over finite sets) with Property (SDC).Primitive positive clones seem to be of particular interest, for the following reason.For a primitive positive clone P ≤ O A , let us consider the set C ( P ) = { C ≤ O A : C ∗ = P } . The greatest element of this set is P ∗ , since C ∗ = P implies that C ⊆ C ∗∗ = P ∗ and P ∗ ∈ C ( P ) follows from P ∗∗ = P . If a clone C ∈ C ( P ) has Property (SDC), thenevery set T ⊆ A n that is closed under C ∗ = P arises as the solution set of a system E of equations over C . Since C ⊆ P ∗ , we can regard E as a system of equations over P ∗ .Therefore, every set T ⊆ A n that is closed under ( P ∗ ) ∗ = P arises as the solution setof a system of equations over P ∗ , i.e., P ∗ has Property (SDC). Thus if P ∗ does notsatisfy Property (SDC), then no clone in C ( P ) can have Property (SDC). In otherwords, primitive positive clones have the “highest chance” for having Property (SDC).Another topic worth further study is the relationship with homomorphism-homo-geneity. It was proved in [9] that homomorphism-homogeneity is equivalent to acertain quantifier elimination property (but somewhat different from Theorem 3.6).Also, our results together with [4] imply that all finite lattices and semilattices withProperty (SDC) are homomorphism-homogeneous, so it might be plausible that Prop-erty (SDC) implies homomorphism-homogeneity in general for finite algebras. Acknowledgments
The authors are grateful to Mikl´os Mar´oti, Dragan Maˇsulovi´c and L´aszl´o Z´adorifor helpful discussions.Research partially supported by the Hungarian Research, Development and Inno-vation Office under grants KH126581 and K128042 and by the Ministry of HumanCapacities, Hungary grant 20391-3/2018/FEKUSTRAT.Open Access. FundRef: University of Szeged Open Access Fund, Grant number:4466.
Appendix: figures and tables for the proof of Theorem 5.3 c b a u c b a u Figure 3.
The elements a, b and c (with an example u proving( a, c, b ) , ( b, a, c ) ∈ T ) in case ( N ). ba c u u cba Figure 4.
The elements a, b and c (with an example u proving( a, b, c ) , ( a, c, b ) ∈ T ) in case ( M ). OLUTION SETS OF SYSTEMS OF EQUATIONS OVER FINITE (SEMI)LATTICES 15 = x y z x ∧ y x ∧ z y ∧ z x ∧ y ∧ zx − ( a, c, b ) ( a, c, b ) ( a, c, b ) ( b, a, c ) ( a, c, b ) ( a, c, b ) y − − ( a, c, b ) ( a, c, b ) ( a, c, b ) ( a, c, b ) ( a, c, b ) z − − − ( a, c, b ) ( a, c, b ) ( a, c, b ) ( a, c, b ) x ∧ y − − − − ( a, c, b ) ( b, a, c ) ( b, a, c ) x ∧ z − − − − − ( a, c, b ) ( a, c, b ) y ∧ z − − − − − − x ∧ y ∧ z − − − − − − − Table 2.
Counterexamples for case ( N ) showing that these equa-tions do not belong to Eq( T ).= x y z x ∧ y x ∧ z y ∧ z x ∧ y ∧ zx − ( a, b, c ) ( a, b, c ) ( a, b, c ) ( a, c, b ) ( a, b, c ) ( a, b, c ) y − − ( a, b, c ) ( a, c, b ) ( a, b, c ) ( a, c, b ) ( a, c, b ) z − − − ( a, b, c ) ( a, b, c ) ( a, b, c ) ( a, b, c ) x ∧ y − − − − ( a, b, c ) ( a, c, b ) ( a, c, b ) x ∧ z − − − − − ( a, b, c ) ( a, b, c ) y ∧ z − − − − − − x ∧ y ∧ z − − − − − − − Table 3.
Counterexamples for case ( M ) showing that these equa-tions do not belong to Eq( T ). z y x x z V V x y z x z Figure 5. ( x , y , z ) satisfies all equations in Eq( T ), but ( x , y , z ) / ∈ T . References [1] V. G. Bodnarˇcuk, L. A. Kaluˇznin, V. N. Kotov, B. A. Romov,
Galois theory for Post algebrasI-II , Kibernetika (Kiev) (1969), 1–10; (1969), 1–9. (Russian) Translated in Cybernet. SystemsAnal. (1969), 243–252; (1969), 531–539.[2] S. Burris, R. Willard, Finitely many primitive positive clones , Proc. Amer. Math. Soc. (1987),no. 3, 427–430.[3] A. Di Nola, G. Lenzi, G. Vitale,
Algebraic geometry for (cid:96) -groups , Algebra Universalis (2018)64.[4] I. Dolinka, D. Maˇsulovi´c, Remarks on homomorphism-homogeneous lattices and semilattices ,Monatsh. Math. (2011), no. 1, 23–37.[5] D. Geiger,
Closed systems of functions and predicates , Pacific J. Math. (1968), 95–100.[6] G. Gr¨atzer, Boolean functions on distributive lattices , Acta Math. Acad. Sci. Hungar. (1964),195-201.[7] G. Gr¨atzer, Lattice Theory: Foundation , Birkh¨auser, 2011. [8] Ju. I. Janov, A. A. Muˇcnik,
Existence of k -valued closed classes without a finite basis , Dokl.Akad. Nauk SSSR (1959), 44–46. (in Russian).[9] D. Maˇsulovi´c, M. Pech, Oligomorphic transformation monoids and homomorphism-homogeneousstructures , Fund. Math. (2011), no. 1, 17–34.[10] B. I. Plotkin,
Some Results and Problems Related to Universal Algebraic Geometry , Internat.J. Algebra Comput. (2007), no. 5-6, 1133–1164.[11] E. L. Post, The two-valued iterative systems of mathematical logic , Annals of MathematicsStudies, no. 5, Princeton University Press, Princeton, N. J., 1941.[12] E. T´oth, T. Waldhauser,
On the shape of solution sets of systems of (functional) equations ,Aequationes Math. (2017), no. 5, 837–857.(E. T´oth) Bolyai Institute, University of Szeged, Aradi v´ertan´uk tere 1, H–6720 Szeged,Hungary
E-mail address : [email protected] (T. Waldhauser) Bolyai Institute, University of Szeged, Aradi v´ertan´uk tere 1, H–6720Szeged, Hungary
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