Building relativized representations using games
aa r X i v : . [ m a t h . L O ] A p r The Andreka-Resek-Thompson and Ferenczi’sresults using games
Mohamed Khaled and Tarek Sayed AhmedDepartment of Mathematics, Faculty of Science,Cairo University, Giza, Egypt.
Abstract
We provide a new proof of the celebrated Andr´eka-Resek-Thompsonrepresentability result of certain finitely axiomtaized cylindric-like alge-bras, together with its quasipolyadic equality analogue proved by Ferenzci.Our proof uses games as introduced in algebraic logic by Hirsch and Hod-kinson. Using the same method, we prove a new representability resultfor diagaonal free reducts of such algebras. Such representability resultsprovides completeness theorems for variants of first order logic, that canalso be viewed as multi-modal logics. Finally, using a result of Marx, weshow that all varieties considered enjoy the superamalgmation property, astrong form of amalgamation, implying that such logics also enjoy a Craiginterpolation theorem.
Stone’s representation theorem for Boolean algebras can be formulated in two,essentially equivalent ways. Every Boolean algebra is isomorphic to a fieldof sets, or the class of Boolean set algebras can be axiomatized by a finiteset of equations. As is well known, Boolean algebras constitute the algebraiccounterpart of propositional logic. Stone’s representation theorem, on the otherhand, is the algebraic equivalent of the completeness theorem for propositionallogic.However, when we step inside the realm of first order logic, things tend tobecome more complicated. Not every abstract cylindric algebra is representableas a field of sets, where the extra Boolean operations of cylindrifiers and diagonalelements are faithfully represented by projections and equality. Disappointingly,the class of representable algebras fail to be axiomatized by any reasonablefinite schema and its resistance to such axiomatizations is inevitable. This isbasically a reflection of the essential incompleteness of natural (more basic)infinitary extensions of first order logic. In such extensions, unlike first orderlogic, validity cannot be captured by a finite schema.Such extentions are obtained by dropping the condition of local finiteness(reflecting the simple fact that first order formulas contain only finitely manyvariables) in algebras considered, allowing formulas of infinite length. This isnecessary if we want to deal with the so-called algebriasable extensions of firstorder logic; extensions that are akin to universal algebraic investigations.1he condition of local finiteness, cannot be expressed in first order logic, andthis is not warranted if we want to deal, like in the case of Boolean algebras,only with equations, or at worst quasi-equations. Then we are faced with thefollowing problem. Find a simple (hopefully finitary) axiomatization of classesof representable algebras abounding in algebraic logic, using only equationsor quasi equations, which also means that we want to stay in the realm ofquasivarieties.There are two conflicting but complementary facets of such a problem, re-ferred to in the literature, as the representation problem. One is to delve deeplyin investigating the complexity of potential axiomatizations for existing vari-eties of representable algebras, the other is to try to sidestep such wild unrulycomplex axiomatizations, often referred to as taming methods . Those tamingmethods can either involve passing to (better behaved) expansions of the alge-bras considered, or else change the very notion of representatiblity involved, aslong as it remains concrete enough. The borderlines are difficult to draw, wedo might not know what is not concrete enough, but we can judge that a givenrepresentability notion is satisfactory, once we have one. (This is analogous toundecidability issues, with the main difference that we do know what we meanby not decidable . We do not have an analogue of a ’recursive representabilitynotion’).One of the taming methods is relativization , meaning that we search for rep-resentations on sets consisting of arbitrary α sequences( α an ordinal specifyingthe dimension of algebras considered), rather than squares, that is set of theform α U for some set U . It turns out that this can be done when do not in-sist on commutativity of cylindrifications. Dropping commutativity makes lifemuch easier in many respects, not only representability. An example is decid-ability of the equational theory of the class of algebras in question. Typicallygiven a set of equations Σ, show that A | = Σ iff A is representable as an algebrawhose elements are genuine relations and operations are set theoretic operationspending only on manipulations of concrete relations. Very few positive resultsare known in this regard, the most famous is the Resek - Thompson celebratedtheorem proved in [1]. The Resek - Thompson result is a refinement of a resultof Resek due to Thompson. The first proof of Resek’s result (that is slightlydifferent) was more than 100 pages long. The short proof of the modified Resek- Thompson result in [1] is due to Andr´eka. In this paper we provide also arelatively short proof of this theorem using games as introduced in algebraiclogic by Hirsch and Hodkinson [2].In [7], Ferenczi talks about an important case for fields of sets that occurwhen the unit consists of a certain set of α -sequences. In addition to the usualset theoretic boolean operations, the i th cylindrification and the constants ij thdiagonal, new natural operations are imposed to describe such field of sets. Suchoperations are e.g. the elementary substitution [ i | j ] and the elementary trans-position [ i, j ] for every i, j < α , restricted to the unit. Ferenczi considers theextended field of sets which is closed under these operations and then gives apositive answer to the question: Do these fields of sets form a variety, and if so,what is its axiomatization?Again, in this paper we provide a shorter proof of Ferenczi’s result, which pro-vides a finite axiomatization, using games as introduced in [2]. We follow theaxiomatization provided by Ferenczi in his recent paper [7]. We refer the readerto [7] and [1] to get a grasp of the importance of this problem in algebraic logic.2uilding representations can be implemented by the step-by-step method(as in [1] and [7]), which consists of treating defects one by one and then takinga limit where the contradictions disappear. What can be done by step-by-stepconstructions, can be done by games but not the other way round. Games wereintroduced in algebraic logic by Hirsch and Hodkinson. Such games, which arebasically Banach-Mazur games in disguise, are games of infinite lengths betweentwo players ∀ and ∃ . The real advantage of the game technique is that games donot only build representations, when we know that such representations exist,but they also tell us when such representations exist, if we do not know a priorithat they do. The translation however from step-by step techniques to gamesis not always a purely mechanical process, even if we know that it can be done.This transfer can well involve some ingenuity, in obtaining games are transpar-ent, intuitive and easy to grasp. It is an unsettled (philosophical) question asto which is more intuitive, step-by-step techniques or games. Basically this de-pends on the context, but in all cases it is nice to have both available if possible,when we know one exists. When we have a step-by-step technique, then we aresure that there is at least one corresponding game. Choosing a simple game iswhat counts at the end.We follow the notation and terminology of [1] and [7]. In particular, forrelations R, S R | S = { ( a, b ) : ∃ c [( a, c ) ∈ R, ( c, b ) ∈ S ] } . Definition 1.1. Class
Crs α An algebra A is a cylindric relativised set algebraof dimension α with unit V if it is of the form h A , ∪ , ∩ , ∼ V , φ, V , C V i , D V ij i i,j<α , where V is a set of α -termed sequences such that A is a non-empty setof subsets of V , closed under the Boolean operations ∪ , ∩ , ∼ V and underthe cylindrifications C V i X = { y ∈ V : y iu ∈ X for some u } , where i < α , X ∈ A , and A contain the elements φ , V and the diagonals D V ij = { y ∈ V : y i = y j } . Class
Drs α An algebra A is a diagonal free relativised set algebra of dimension α with unit V if it is of the form h A , ∪ , ∩ , ∼ V , φ, V , C V i i i,j<α , where V is a set of α -termed sequences such that A is a non-empty setof subsets of V , closed under the Boolean operations ∪ , ∩ , ∼ V and underthe cylindrifications C V i X = { y ∈ V : y iu ∈ X for some u } , where i < α , X ∈ A , and A contain the elements φ and V . The meaning of the notation y iu is ( y iu ) j = y j if j = i and ( y iu ) j = u if j = i . Some concepts and notation concerning
Crs α and Drs α : . - The class Crs α is a subclass of Drs α .- Let A ∈ Crs α be with unit element V . The relativized substitution operator V S ij is defined as V S ij X = C V i ( D V ij ∩ X ) ( X ∈ A ) . We often omit the superscript V from V S ij and write S ij . Clearly A is closed under V S ij , since A is closed under the cylindrifications and containsthe diagonals. The transformation τ defined on α is called finite if τ i = i except for finitelymany i < α . Important special cases of the finite transformations are thetransformations [ i | j ] , called elementary substitutions, and [ i, j ] , called trans-positions.- Let A ∈ Drs α be with unit element V . If y ∈ V and τ is any finite trans-formation on α , V S τ y is defined as τ | y . If X ∈ A , then V S τ X is defined as { τ | y : y ∈ X } . Definition 1.2 (Class D α ) . It is the subclass of
Crs α for which V S ij V = V forevery i, j < α , where V is the unit of the algebra. If we consider the substitution operators V S [ i | j ] and the transposition operators V S [ i,j ] defined above, then we get the polyadic versions of Crs α . Definition 1.3. Class
P rs α An algebra B = h A , ∪ , ∩ , ∼ V , φ, V , C V i , V S [ i | j ] , V S [ i,j ] i i,j<α is an α -dimensional polyadic relativized set algebra if for the diagonal freereduct, Rd df B ∈ Drs α ; further, A is closed under the transpositions V S [ i,j ] and the substitutions V S [ i | j ] . Class
P ers α An algebra B = h A , ∪ , ∩ , ∼ V , φ, V , C V i , V S [ i | j ] , V S [ i,j ] , D V ij i i,j<α is an α -dimensional polyadic equality relativized set algebra if for the cylin-dric reduct, Rd ca B ∈ Crs α ; further, A is closed under the transpositions V S [ i,j ]3 . Class
Srs α An algebra B = h A , ∪ , ∩ , ∼ V , φ, V , C V i , V S [ i | j ] i i,j<α is an α -dimensional substitution relativized set algebra if for the diagonalfree reduct, Rd df B ∈ Drs α ; further, A is closed under the substitutions V S [ i | j ] . Definition 1.4 (Class Dp α ) . It is the subclass of
P rs α for which V S [ i | j ] V = V for every i, j < α , where V is the unit of the algebra. Definition 1.5 (Class
Dpe α ) . It is the subclass of
P ers α for which V S [ i | j ] V = V for every i, j < α , where V is the unit of the algebra. Definition 1.6 (Class Ds α ) . It is the subclass of
Srs α for which V S [ i | j ] V = V for every i, j < α , where V is the unit of the algebra. We assume the knowledge of the concepts of cylindric algebras [3]. Thecylindric axiom( C ) c i c j x = c j c i x A need not to be closed under V S τ for any arbitrary τ . Clearly, A is closed under the substitutions V S [ i | j ] because V S [ i | j ] = V S ij . MGR ) are postulated. By Resek-Thompson theorem [1], theexistence of such axioms yields representability by relativized set algebra. Thefollowing axiomatization is due to Thompson and Andreka.
Definition 1.7 (Class
P T A α ) . An algebra A = h A, + , · , − , , , c i , d ij i i,j ∈ α ,where + , · are binary operations, − , c i are unary operations and , , d ij areconstants for every i, j ∈ α , is partial transposition algebra if it satisfies thefollowing identities for every i, j, k ∈ α . ( C ) − ( C ) h A, + , · , − , , , c i i i ∈ α is a Boolean algebra with additive closure op-erators c i such that the complements of c i -closed elements are c i -closed, ( C ) ∗ c i c j x ≥ c j c i x · d jk if k / ∈ { i, j } , ( C ) d ii = 1 , ( C ) d ij = c k ( d ik · d kj ) if k / ∈ { i, j } , ( C ) d ij · c i ( d ij · x ) ≤ x if i = j , (MGR) for every i, j ∈ α , i = j , let s ij x = c i ( d ij · x ) , s ii x = x . Then: s ki s ij s jm s mk c k x = s km s mi s ij s jk c k x if k / ∈ { i, j, m } , m / ∈ { i, j } . We also assume the basic knowledge of the concepts of polyadic equalityand quasi-polyadic equality algebras [4, p. 266]. The so called finitary polyadicequality algebras, i.e., the class
F P EA α , is term defnitionally equivalent tothe quasi-polyadic equality algebras [10, Theorem 1]. The axiomatization of F P EA α and of the class T EA α to be introduced are different in only oneaxiom, namely the axiom ( F ) s ij c k x = c k s ij x if k / ∈ { i, j } .The following axiomatization is due to Ferenczi, abstracting away from theclass P ers α (meaning that the axioms all hold in P ers α ). This is a soundnesscondition. Ferenzci proves completeness of these axioms, which we also proveusing the different technique of resorting to games. Definition 1.8 (Class
T EA α ) . A transposition equality algebra of dimension α is an algebra A = h A, + , · , − , , , c i , s ij , s ij , d ij i i,j ∈ α , where c i , s ij , s ij are unary operations, d ij are constants, the axioms ( F )-( F )below are valid for every i, j, k < α : ( F e ) h A, + , · , − , , i is a boolean algebra, s ii = s ii = d ii = Id ↾ A and s ij = s ji , ( F e ) x ≤ c i x , We call it partial transposition algebra to illustrate that MGR axiom gives partial trans-positions. This class is different from the class of partial transposition algebras defined in[6]. F e ) c i ( x + y ) = c i x + c i y , ( F e ) s ij c i x = c i x , ( F e ) c i s ij x = s ij x , i = j , ( F e ) ∗ s ij s km x = s km s ij x if i, j / ∈ { k, m } , ( F e ) s ij and s ij are boolean endomorphisms, ( F e ) s ij s ij x = x , ( F e ) s ij s ik x = s jk s ij x , i, j, k are distinct, ( F e ) s ij s ij x = s ji x , ( F e ) s ij d ij = 1 , ( F e ) x · d ij ≤ s ij x .For A ∈ T A α , its partial transposition reduct is the structure Rd pt A = h A, + , · , − , , , c i , s ij , d ij i i,j ∈ α . The following axiomatization is new, it is obtained from Ferenczi’s axioma-tization by dropping equations involving diagonal elements.
Definition 1.9 (Class
T A α ) . A transposition algebra of dimension α is analgebra A = h A, + , · , − , , , c i , s ij , s ij i i,j ∈ α , where c i , s ij , s ij are unary operations, the axioms ( F )-( F ) below are valid forevery i, j, k < α : ( F ) h A, + , · , − , , i is a boolean algebra, s ii = s ii = d ii = Id ↾ A and s ij = s ji , ( F ) x ≤ c i x , ( F ) c i ( x + y ) = c i x + c i y , ( F ) s ij c i x = c i x , ( F ) c i s ij x = s ij x , i = j , ( F ) ∗ s ij s km x = s km s ij x if i, j / ∈ { k, m } , ( F ) s ij and s ij are boolean endomorphisms, ( F ) s ij s ij x = x , ( F ) s ij s ik x = s jk s ij x , i, j, k are distinct, ( F ) s ij s ij x = s ji x . The following axiomatization is new. It is similar to Pinter’s axiomatization[12] with two major differences. We do not have commutativity of cylindrifica-tions, this is one thing; the other is that we stipulate the
M GR identities.6 efinition 1.10 (Class SA α ) . A substitution algebra of dimension α is analgebra A = h A, + , · , − , , , c i , s ij i i,j ∈ α , where c i , s ij are unary operations, the axioms ( S )-( F ) below are valid for every i, j, k < α : ( S ) h A, + , · , − , , i is a boolean algebra and s ii = Id ↾ A , ( S ) x ≤ c i x , ( S ) c i ( x + y ) = c i x + c i y , ( S ) s ij c i x = c i x , ( S ) c i s ij x = s ij x , i = j , ( S ) ∗ s ij s km x = s km s ij x if i, j / ∈ { k, m } , ( S ) s ij is boolean endomorphism, ( S ) s kk s jk x = s ki s ji x , ( S ) s ki s ij s jm s mk c k x = s km s mi s ij s jk c k x if k / ∈ { i, j, m } , m / ∈ { i, j } . ( F ) ∗ (and also ( F ) ∗ and ( S ) ∗ ) is obviously a weakening of ( F ). Also itis known that Rd pt A ∈ P T A α , for any A ∈ T EA α [10]. We consider as knownthe concept of the substitution operator s τ defined for any finite transformation τ on α ; s τ can be introduced uniquely in F P EA α and in T EA α , too. Theexistence of such an s τ follows from the proof of [10, Theorem 1(ii)], it is easy tocheck that the proof works by assuming ( F ) ∗ instead of ( F ) and (notationally)the composition operator | instead of ◦ .Throughout this paper we assume that the polyadic-like algebras occurringhere are equipped with the operator s τ , where τ is finite. Further, s τ is assumedto have the following properties for arbitrary finite transformations τ and λ andordinals i, j < α (by [10, p.542]): s τ | λ = s τ s λ , s ij = s [ i | j ] , s τ d ij = d τiτj (of course only in the class T EA α ), c i s τ ≤ s τ c τ − , here τ is finite permutation. Lemma 1.1.
Let α be an ordinal, A ∈ T EA α , a ∈ At A and i, j ∈ α . Then a ≤ d ij = ⇒ s [ i,j ] a = a. Proof.
See [7, p. 875]. This depends on ( F ) and ( F ). Games and Networks
In this section fix n ∈ ω . We start with some preparations. Let x , y be n -tuplesof elements of some set. We write x i for the ith element of x , for i < n , so that x = ( x , · · · , x n − ). For i < n , we write x ≡ i y if x j = y j for all j < n with j = i . The next two definitions are taken from [2]. A (relativized) network is afinite approximation to a (relativized) representation. Definition 2.1. • Let A ∈ P T A n . A relativized A pre-network is a pair N = ( N , N ) where N is a finite set of nodes N : N n → A is a partial map, suchthat if f ∈ domN , and i, j < n then f if ( j ) ∈ DomN . N is atomic if RangeN ⊆ At A . We write N for any of N, N , N relying on context,we write nodes ( N ) for N and edges ( N ) for dom ( N ) . N is said to be anetwork if(a) for all ¯ x ∈ edges ( N ) , we have N (¯ x ) ≤ d ij iff x i = x j .(b) if ¯ x ≡ i ¯ y , then N (¯ x ) · c i N (¯ y ) = 0 . • Let A ∈ T EA n . A relativized A pre-network is a pair N = ( N , N ) where N is a finite set of nodes N : N n → A is a partial map, such that if f ∈ domN , and τ is a finite transformation then τ | f ∈ DomN . Again N is atomic if RangeN ⊆ At A . Also we write N for any of N, N , N relying on context, we write nodes ( N ) for N and edges ( N ) for dom ( N ) . N is said to be a network if(a) for all ¯ x ∈ edges ( N ) , we have N (¯ x ) ≤ d ij iff x i = x j ,(b) if ¯ x, ¯ y ∈ edges ( N ) and ¯ x ≡ i ¯ y , then N (¯ x ) · c i N (¯ y ) = 0 ,(c) N ([ i, j ] | ¯ x ) = s [ i,j ] N (¯ x ) , for all ¯ x ∈ edges ( N ) and all i, j < n . Definition 2.2.
Let A ∈ P T A n ∪ T EA n . We define a game denoted by G ω ( A ) with ω rounds, in which the players ∀ (male) and ∃ (female) build an infinitechain of relativized A pre-networks ∅ = N ⊆ N ⊆ . . . . In round t , t < ω , assume that N t is the current prenetwork, the players moveas follows:(a) ∀ chooses a non-zero element a ∈ A , ∃ must respond with a relativizedprenetwork N t +1 ⊇ N t containing an edge e with N t +1 ( e ) ≤ a ,(b) ∀ chooses an edge ¯ x of N t and an element a ∈ A . ∃ must respond with apre-network N t +1 ⊇ N t such that either N t +1 (¯ x ) ≤ a or N t +1 (¯ x ) ≤ − a ,(c) or ∀ may choose an edge ¯ x of N t an index i < n and b ∈ A with N t (¯ x ) ≤ c i b . ∃ must respond with a prenetwork N t +1 ⊇ N t such that for some z ∈ N t +1 ,N t +1 (¯ x iz ) = b . ∃ wins if each relativized pre-network N , N , . . . played during the game is ac-tually a relativized network. Otherwise, ∀ wins. There are no draws. Lemma 2.1.
Let A ∈ P T A n be atomic. For all i, j ∈ n , i = j , define t ij x = d ij · c i x and t ii x = x . Then(i) ( t ij ) A : At A → At A (ii) Let Ω = { t ji : i, j ∈ n } ∗ , where for any set H , H ∗ denotes the free monoidgenerated by H . Let σ = t i j . . . t i n j n be a word. Then define for a ∈ A : σ A ( a ) = ( t i j ) A (( t i j ) A . . . ( t i n j n ) A ( a ) . . . ) , and ˆ σ = [ i | j ] | [ i | j ] . . . | [ i n | j n ] . Then A | = σ ( x ) = τ ( x ) if ˆ σ = ˆ τ , σ, τ ∈ Ω . That is for all σ, τ ∈ Ω , if ˆ σ = ˆ τ , then for all a ∈ A , we have σ A ( a ) = τ A ( a ) .Proof. cf. [1] proof of Lemma 1 therein. The MGR , merry go round identitiesare used here. We note that Andreka’s proof of this lemma is long, but usingfairly obvious results on semigroups a much shorter proof can be given.
Lemma 2.2.
Let A ∈ P T A n be atomic. For all i, j ∈ n , i = j , let t ij x be asabove. The following hold for all i, j, k, l ∈ n :(i) ( t ij ) A x ≤ d ij for all x ∈ A .(ii) ( x ≤ d ij ⇒ ( t ki ) A x ≤ d ij · d ik · d jk ) for all x ∈ A .(iii) ( a ≤ c i b ⇔ c i a = c i b ) for all a, b ∈ At A .(iv) c i ( t ij ) A x = c i x for all x ∈ A .Proof. ( i ) Follows directly from the definition of ( t ij ) A .( ii ) First we need to check the following for all i, j, k < n :- c k d ij = d ij if k / ∈ { i, j } . For, see [3, Theorem 1.3.3], the proof doesn’tinvolve ( C ).- d ij = d ji . For, see [3, Theorem 1.3.1], the proof works, indeed itdoesn’t depend on ( C ).- d ij · d jk = d ij · d ik . A proof for such can be founded in [3, Theorem1.3.7]. 9ow we can proof ( ii ): t ki x = d ik · c k x ≤ d ik · c k d ij = d ik · d ij (1)= d ki · d ij = d ki · d kj (2)From (1), (2) the desired follows.( iii ) − ( iv ) See [1, p. 675-676] Definition 2.3 (Partial transposition network) . Let A be an atomic P T A n and fix an atom a ∈ At A . Let ¯ x be any n -tuple (of nodes) such that x i = x j ifand only if a ≤ d ij for all i, j < n . Let N SQ ¯ x = { ¯ y ∈ n { x , x , · · · , x n − } : | Range (¯ y ) | < n } ( N S stands for non-surjective sequences). We define the par-tial transposition network
P T ( a )¯ x : N SQ ¯ x → At A As follows: If ¯ y ∈ N SQ ¯ x , then ¯ y = [ i | j ] | [ i | j ] | · · · | [ i k | j k ] | ¯ x , for some i , i , · · · , i k , j , j , · · · , j k < n . Let P T ( a )¯ x (¯ y ) = ( t j i · · · t j k i k ) ( A ) a . This is well defined by Lemma 2.1 Definition 2.4 (Transposition network) . Let A be an atomic T EA n and fix anatom a ∈ At A . Let ¯ x be any n -tuple of nodes such that x i = x j if and only if a ≤ d ij for all i, j < n . Let Q ¯ x = n { x , x , · · · , x n − } . Consider the followingequivalence relation ∼ on Q ¯ x : ¯ y ∼ ¯ z if and only if ¯ z = τ | ¯ y for some finite permutation τ, ¯ y, ¯ z ∈ Q ¯ x .Let us choose and fix representative tuples for the equivalence classes concerning ∼ such that each representative tuple is of the form [ i | j ] | [ i | j ] | · · · | [ i k | j k ] | ¯ x for some k ≥ , i , i , · · · , i k , j , j , · · · , j k < n . Such representative tuplesexist. Indeed, for every ¯ y ∈ Q ¯ x , ∃ τ finite permutation, ∃ k ≥ , i , i , · · · , i k , j , j , · · · , j k < n such that ¯ y = τ | [ i | j ] | [ i | j ] | · · · | [ i k | j k ] | ¯ x. Let ¯ z = τ − | ¯ y , then ¯ z ∼ ¯ y and ¯ z = [ i | j ] | [ i | j ] | · · · | [ i k | j k ] | ¯ x . Let Rt de-note this fixed set of representative tuples. We define the transposition network T ( a )¯ x : Q ¯ x → At A as follows: • If ¯ y ∈ Rt , then ¯ y = [ i | j ] | [ i | j ] | · · · | [ i k | j k ] | ¯ x , for some i , i , · · · , i k , j , j , · · · , j k < n . Let T ( a )¯ x (¯ y ) = ( t j i · · · t j k i k ) ( Rd pt A ) a . This is well definedby Lemma 2.1. • If ¯ z = σ | ¯ y for some finite permutation σ and some ¯ y ∈ Rt , then let T ( a )¯ x (¯ z ) = s σ T ( a )¯ x (¯ y ) . Lemma 2.3.
The above definition is unique. roof. The first part is well defined by Lemma 2.1. Now we need to provethat if σ | ¯ y = τ | ¯ y for some finite permutations σ, τ and some ¯ y ∈ Rt , then s σ T ( a )¯ x (¯ y ) = s τ T ( a )¯ x (¯ y ). First, we need the following: Claim . If ¯ y = τ | ¯ y for some finite permutation τ and some ¯ y ∈ Rt , then T ( a )¯ x (¯ y ) = s τ T ( a )¯ x (¯ y ) . Proof.
It suffices to show that if ¯ y = [ i, j ] | ¯ y for some i, j < n and some ¯ y ∈ Rt ,then T ( a )¯ x (¯ y ) = s [ i,j ] T ( a )¯ x (¯ y ). For, suppose that ¯ y = [ i, j ] | ¯ y for some i, j < n andsome ¯ y ∈ Rt , then y i = y j and then T ( a )¯ x (¯ y ) ≤ d ij by Lemma 2.2. Hence byLemma 1.1, T ( a )¯ x (¯ y ) = s [ i,j ] T ( a )¯ x (¯ y ).Returning to our prove, assume that σ | ¯ y = τ | ¯ y for some finite permutations σ, τ and some ¯ y ∈ Rt . Then ( τ − | σ ) | ¯ y = ¯ y and so s ( τ − | σ ) T ( a )¯ x (¯ y ) = T ( a )¯ x (¯ y ).Therefore, s τ − s σ T ( a )¯ x (¯ y ) = T ( a )¯ x (¯ y ). Hence, s σ T ( a )¯ x (¯ y ) = s τ T ( a )¯ x (¯ y ), and we aredone. Lemma 2.4.
1. Let A be an atomic P T A n and a ∈ At A . Let ¯ x be any n -tuple of nodessuch that x i = x j if and only if a ≤ d ij for all i, j < n . Then P T ( a )¯ x is anatomic A network.2. Let A be an atomic T EA n and a ∈ At A . Let ¯ x be any n -tuple of nodessuch that x i = x j if and only if a ≤ d ij for all i, j < n . Then T ( a )¯ x is anatomic A network.Proof. Straightforward from the above .
Lemma 2.5.
Let A ∈ P T A n ∪ T EA n . Then ∃ can win any play of G ω ( A ) .Proof. The proof is similar to that of Lemma 7.8 in [2]. Let A + be the canonicalextension of A . First note that A + ∈ T EA n and A + is atomic. Of course any A pre-network is an A + pre-network. In each round t of the game G ω ( A ), where N t is as above, ∃ constructs an atomic A + network M t satisfying M t ⊇ N t , nodes ( M t ) = nodes ( N t ) , edges ( M t ) = edges ( N t ) . Then if ¯ x ≡ i ¯ y , we have N t (¯ x ) · c i N t (¯ y ) ≥ M t (¯ x ) · c i M t (¯ y ) = 0 . ∃ starts by M = N = ∅ . Suppose that we are in round t and assume inductivelythat ∃ has managed to construct M t ⊇ N t as indicated above. We consider thepossible moves of ∀ .(1) Suppose that ∀ picks a non zero element a ∈ A . ∃ chooses an atom a − ∈ A + with a − ≤ a . She chooses new nodes x , . . . x n − with x i = x j iff a − ≤ d ij . If A ∈ P T A n . She creates two new relativized networks N t +1 , M t +1 withnodes those of N t plus x , . . . x n − and hyperedges those of N t togetherwith N SQ ¯ x . The new hyper labels in M t +1 are defined as follows: M t +1 = M t ∪ P T ( a − )¯ x .
11y Lemma 2.4 it follows that M t +1 is an atomic A + network. Labelsin N t +1 are given by • N t +1 (¯ x ) = a · Q i,j : x i = x j d ij . • N t +1 (¯ y ) = Q i,j : y i = y j d ij for any other hyperedge ¯ y . If A ∈ T EA n . She creates two new relativized networks N t +1 , M t +1 withnodes those of N t plus x , . . . x n − and hyperedges those of N t togetherwith Q ¯ x . The new hyper labels in M t +1 are defined as follows: M t +1 = M t ∪ T ( a − )¯ x . By Lemma 2.4 it follows that M t +1 is an atomic A + network. Labelsin N t +1 are given by • N t +1 ( τ | ¯ x ) = s τ a · Q i,j : x i = x j d τiτj for any finite permutation τ . • N t +1 (¯ y ) = Q i,j : y i = y j d ij for any other hyperedge ¯ y . ∃ responds to ∀ ’s move in round t with N t +1 . One can check that N t ⊆ N t +1 ⊆ M t +1 , as required.(2) If ∀ picks an edge ¯ x of N t and an element a ∈ A , ∃ lets M t +1 = M t and lets N t +1 be the same as N t except that If A ∈ P T A n . N t +1 (¯ x ) = N t (¯ x ) · a if M t (¯ x ) ≤ a and N t +1 (¯ x ) = N t (¯ x ) . − a otherwise. Because M t (¯ x ) is an atom in A + , it follows that if M t (¯ x ) (cid:2) a , then M t (¯ x ) ≤ − a , so this is satisfactory. If A ∈ T EA n . for every finite permutation τ , N t +1 ( τ | ¯ x ) = N t ( τ | ¯ x ) · s τ a if M t ( τ | ¯ x ) ≤ s τ a and N t +1 ( τ | ¯ x ) = N t ( τ | ¯ x ) . − s τ a otherwise. Because M t ( τ | ¯ x ) is an atom in A + for every finite permutation τ , it followsthat if M t ( τ | ¯ x ) (cid:2) s τ a , then M t ( τ | ¯ x ) ≤ − s τ a , so this is satisfactory.(3) Alternatively ∀ picks ¯ x ∈ N t i < n and b ∈ A such that N t (¯ x ) ≤ c i b . Let M t (¯ x ) = a − . If there is z ∈ M t with M t (¯ x iz ) ≤ b then we are done. In moredetail, ∃ lets M t +1 = M t and define N t +1 accordingly. Else, there is no such z . We have c i a − · b = 0 (inside A + ), indeed c i ( c i a − · b ) = c i a − · c i b = c i ( a − · c i b )= c i a − = 0 . (since a − ≤ c i b )Choose an atom b − ∈ A + with b − ≤ c i a − · b . Then we have b − ≤ b and b − ≤ c i a − . But by Lemma 2.2 ( iv ) we also have a − ≤ c i b − . If A ∈ P T A n . Let G be the A + network with nodes { x , . . . x n − , z } , z anew node, and the hyperedges are the sequences in N SQ ¯ x ∪ N SQ ¯ t and G = P T ( a − )¯ x ∪ P T ( b − )¯ t where ¯ t = ¯ x iz . If A ∈ T EA n . Let G be the A + network with nodes { x , . . . x n − , z } , z a new node, and the hyperedges are the sequences in Q ¯ x ∪ Q ¯ t and G = T ( a − )¯ x ∪ T ( b − )¯ t where ¯ t = ¯ x iz .12gain this is well defined. Then, it easy to check that, M t ( i , . . . i n ) = G ( i , . . . i n ) for all i , . . . i n ∈ Range ¯ x . That is the subnetworks of M t and G with nodes Range ¯ x are isomorphic. Then we can amalgamate M t and G and define M t +1 as the outcome. The amalgamation here is possible, sincethe networks are only relativized, we don’t have all hyperedges. By Lemma2.4, one can check that M t +1 as so defined is an atomic A + network. Nowdefine N t +1 accordingly. That is N t +1 has the same nodes and edges of M t +1 , with labelling as for N t except that If A ∈ P T A n . N t +1 (¯ t ) = b . The rest of the other labels are defined to be N t +1 (¯ y ) = Y i,j : y i = y j d ij . If A ∈ T EA n . N t +1 ( τ | ¯ t ) = s τ b for any finite permautation τ . The rest ofthe other labels are defined to be N t +1 (¯ y ) = Y i,j : y i = y j d ij . By Lemma 2.2, it is clear that N t ⊆ N t +1 ⊆ M t +1 and then N t +1 is A network. Hence N t +1 is appropriate to be played by ∃ in response to ∀ ’smove. Theorem 3.1.
Let ≤ n < ω . If A ∈ P T A n , then A ∈ ID n .Proof. Let A ∈ P T A n . We want to build an isomorphism from A to some B ∈ D n . Consider a play N ⊆ N ⊆ . . . of G ω ( A ) in which ∃ plays asin the previous lemma and ∀ plays every possible legal move. The outcomeof the play is essentially a relativized representation of A defined as follows.Let N = S t<ω nodes ( N t ), and edges ( N ) = S t<ω edges ( N t ) ⊆ n N . By thedefinition of the networks, ℘ ( edges ( N )) ∈ D n . We make N into a representationby defining h : A → ℘ ( edges ( N )) as follows h ( a ) = { ¯ x ∈ edges ( N ) : ∃ t < ω (¯ x ∈ N t & N t (¯ x ) ≤ a ) } . ∀ -moves of the second kind guarantee that for any n -tuple ¯ x and any a ∈ A ,for sufficiently large t we have either N t (¯ x ) ≤ a or N t (¯ x ) ≤ − a . This ensuresthat h preserves the boolean operations. ∀ -moves of the third kind ensure thatthe cylindrifications are respected by h . Preserving diagonals follows from thedefinition of networks. The first kind of ∀ -moves tell us that h is one-one. Butthe construction of the game under consideration ensures that h is onto, too. Infact ¯ h is a representation from A onto B ∈ D n . This follows from the definitionof networks. Theorem 3.2.
Let ≤ n < ω . If A ∈ T EA n , then A ∈ IDpe n . roof. Let A ∈ T EA n . As above consider a play N ⊆ N ⊆ . . . of G ω ( A ) inwhich ∃ plays as in the previous lemma and ∀ plays every possible legal move.The outcome of the play is essentially a relativized representation of A definedas follows. Let N = S t<ω nodes ( N t ), and edges ( N ) = S t<ω edges ( N t ) ⊆ n N .Again by the definition of the networks, it is easy to see that ℘ ( edges ( N )) ∈ Dpe n . We make N into a representation by defining h : A → ℘ ( edges ( N )) asfollows h ( a ) = { ¯ x ∈ edges ( N ) : ∃ t < ω (¯ x ∈ N t & N t (¯ x ) ≤ a ) } . As the previous theorem, h preserves the boolean operations, the cylindrifictionsand the diagonals. Now we check transpositions. Let ¯ y ∈ h ( s ij a ). Then thereexists t < ω such that N t (¯ y ) ≤ s ij a . Hence s ij N t (¯ y ) = N t ([ i, j ] | ¯ y ) ≤ a . Theother inclusion is similar. The preservation of the substitutions follows directlyfrom the preservation of the culindrifications and the diagonals. Theorem 3.3.
Let ≤ n < ω . If A ∈ T A n , then A ∈ IDp n .Proof. It is enough to prove that every
T A α is embeddable in a reduct of some T EA α . For, use the same method in the proof of Proposition 9. in [10], ituses only axioms ( F ), ( F ) and ( F ). This method depend on the fact that A is definable by positive equations only. So A is canonical, and we can definediagonals in this canonical extension, B say, by d ij = T { y ∈ B : s ij y = 1 } ,for every i, j ∈ α . Then it can be shown that B with those constants is in T EA α . Theorem 3.4.
Let ≤ n < ω . If A ∈ SA n , then A ∈ IDs n .Proof. We will use the same analogue in the proof of Proposition 9. in [10].Now, let 2 ≤ n < ω and A ∈ SA n . Then A is definable by positive equationsonly. Indeed, There is an axiomatization of Boolean algebra involving onlymeets and joins, so a Boolean homomorphism is specified by respecting meetsand joins. By this we get rid of negation. Therefore by (I) in [3, p. 440], A is a subalgebra of a complete and atomic B such that B | = ( C − C ). Let d ij = T { y ∈ B : s ij y = 1 } , for every i, j ∈ α . This definition is justified because B is complete. Our aim is to prove that B with this constants satisfies theaxioms C − C and this finishes the prove. For, it is enough to prove that B | = ( C − C ). Claim 3.1.
For every i, j ∈ α and every set K , s ij ( T k ∈ K y k ) = T k ∈ K s ij y k .Proof. See [10, Claim 9.1.]
Claim 3.2.
For every i, j ∈ α , B | = s ij d ij = 1 .Proof. Let i, j ∈ α , B | = s ij d ij = s ij { y ∈ B : s ij y = 1 } = { s ij y : y ∈ B and s ij y = 1 } = 1 . Claim 3.3.
For every i, j ∈ α and every x ∈ B , if i = j then B | = s ij x = c i ( x · d ij ) . roof. Let y ∈ B be such that s ij y = 1. B | = s ij [ − ( x · y ) + s ij x ] = − ( s ij x · s ij y ) + s ij s ij x = − s ij x + s ij x = 1 , therefore, B | = x · y ≤ s ij x . Hence, B | = \ { x · y : y ∈ B and s ij y = 1 } ≤ s ij x B | = x · d ij ≤ s ij x, i.e., B | = c i ( x · d ij ) ≤ s ij x . On the other direction, B | = x · d ij ≤ c i ( x · d ij ) B | = s ij ( x · d ij ) ≤ s ij ( c i ( x · d ij )) B | = s ij x · s ij d ij ≤ c i ( x · d ij ) B | = s ij x ≤ c i ( x · d ij ) . Now, B | = d ij = { y ∈ A : s ii y = 1 } = { y ∈ A : y = 1 } = 1 . B | = s ij [ − ( d ij · c i ( d ij · x )) + x ]= − ( s ij d ij · s ij c i ( d ij · x )) + s ij x = − c i ( d ij · x ) + s ij x ≥ − s ij x + s ij x = 1 . Therefore, B | = d ij · c i ( d ij · x ) ≤ x . B | = s ij [ − c k ( d ik · d kj ) + d ij ]= s ij [ − s ki d kj + d ij ]= − s ij s ki d kj + s ij d ij = − s ij s kj d kj + 1= 1 . Therefore, B | = c k ( d ik · d kj ) ≤ d ij . Hence we proved that B | = ( C − C ) andalso B | = s ij x = c i ( d ij · x ). This finishes the prove.15e have proved our theorems for finite dimensions. Now we turn to theinfinite dimensional case. We give a general method of lifting representability forfinite dimensional cases to the transfinite; that can well work in other contexts. Theorem 3.5.
Assume that α ≥ ω . Then P T A α = ID α , T EA α = IDpe α , T A α = IDp α and SA α = IDs α .Proof. We will consider the case of
P T A n and the other cases are similar. First,we know that P T A n = ID n for every finite n < ω . We want to show that P T A α = ID α for any infinite α . First note that :1. For any A ∈ P T A α and ρ : n → α , n ∈ ω and ρ is one to one, define Rd ρ A as in [3, Definition 2.6.1]. Then Rd ρ A ∈ P T A n .2. For any n ≥ ρ : n → α as above, ID n ⊆ S Rd ρ ID α as in [4, Theorem3.1.121].3. ID α is closed under the ultraproducts, cf. [4, Lemma 3.1.90].Now we show that if A ∈ P T A α , then A is representable. First, for any ρ : n → α , Rd ρ A ∈ P T A n . Hence Rd ρ A is in ID n and so it is in S Rd ρ ID α .Let J be the set of all finite one to one sequences with range in α . For ρ ∈ J ,let M ρ = { σ ∈ J : ρ ⊆ σ } . Let U be an ultrafilter of J such that M ρ ∈ U forevery ρ ∈ J . Then for ρ ∈ J , there is B ρ ∈ ID α such that Rd ρ A ⊆ Rd ρ B ρ . Let C = Q B ρ /U ; it is in U pID α = D α . Define f : A → Q B ρ by f ( a ) ρ = a , andfinally define g : A → C by g ( a ) = f ( a ) /U . Then g is an embedding. Remark.
Let A be an algebra in some class of our interest. A is said to becompletely representable if there is a representation f : A → P ( V ) such that S { f ( x ) : x an atom } = V . Therefore, according to the representations thatare built in the proofs of Theorems 3.1, 3.2, 3.3, 3.4 and 3.5, every atomicalgebra in
P T A α ∪ T EA α ∪ T A α ∪ SA α is completely representable. Here we compare our classes with other important classes existing in theliterature. Given a set U and a mapping p ∈ α U , then the set α U ( p ) = { x ∈ α U : x and p are different only in finitely many places } is called the weak space determined by p and U . Definition 3.1. Class Gw α A set algebra in
Crs α is called a generalized weakcylindric relativized set algebra if there are sets U k , k ∈ K , and mappings p k ∈ α U k such that V = S k ∈ K α U ( p k ) k , where V is the unit. Class
Gwp α ( Gwpe α ) A set algebra in
P rs α ( P ers α ) is called a generalizedweak polyadic (equality) relativized set algebra if there are sets U k , k ∈ K ,and mappings p k ∈ α U k such that V = S k ∈ K α U ( p k ) k , where V is the unit. Class
Gws α A set algebra in
Srs α is called a generalized weak substitutionrelativized set algebra if there are sets U k , k ∈ K , and mappings p k ∈ α U k such that V = S k ∈ K α U ( p k ) k , where V is the unit. A characterization of the completely representable algebras, cf, [8, Lemma 2.1]
16 known characterization of the class
Gwpe α is : If V is the unit of an A ∈ P rs α , then A ∈ Gwp α if and only if y ∈ V implies τ | y ∈ V for every finitetransformation τ . Using this property one can prove that Gwpe α = Dpe α , sowe can replace Dpe α by Gwpe α in Theorem 3.2 and Theorem 3.5. But thesame is not true for the other types, for example, for finite n ∈ ω , the class Gw α coincide with the class of locally square cylindric algebras. Andreka gavea finite schema axiomatization for the former class in [9], and Andreka’s axiomsand P T A α are not definitionally equivalent. Definition 4.1.
Let K be a class of algebras having a boolean reduct. A ∈ K is in the amalgamation base of K if for all A , A ∈ K and monomorphisms i : A → A , i : A → A there exist D ∈ K and monomorphisms m : A → D and m : A → D such that m ◦ i = m ◦ i . If in addition, ( ∀ x ∈ A j )( ∀ y ∈ A k )( m j ( x ) ≤ m k ( y ) = ⇒ ( ∃ z ∈ A )( x ≤ i j ( z ) ∧ i k ( z ) ≤ y )) where { j, k } = { , } , then we say that A lies in the super amalgamation baseof K . Here ≤ is the boolean order. K has the (super) amalgamation property (( SU P ) AP ) , if the (super) amalgamation base of K coincides with K . We now show using a result of Marx, that all varieties considered have thesuperamalgmation property (
SU P AP ) . We consider
T A α = ID α The rest of thecases are the same. For a set V , B ( V ) denotes the Boolean algebra ( ℘ ( V ) , ∩ , ∼ ) . Definition 4.2.
1. A frame of type
T A α is a first order structure F = ( V , C i , S ji , S ij ) i,j ∈ α , where V is an arbitrary set and and C i , S ji and S ij are binary relations forall i, j ∈ α .2. Given a frame F , its complex algebra denote by F + is the algebra ( B ( V ) , c i , s ji , s ij ) i,j , where for X ⊆ V , s ji ( X ) = { s ∈ V : ∃ t ∈ X, ( t, s ) ∈ s ji } , and same for c i and s ij .
3. Given K ⊆ T A α , then Cm − K = { F : F + ∈ K } .
4. Given a family ( F i ) i ∈ I a zigzag product of these frames is a substructure of Q i ∈ I F i such that the projection maps restricted to S are onto. Theorem 4.1. (Marx) Assume that K is a canonical variety and L = Cm − K is closed under finite zigzag products. Then K has the superamalgamation prop-erty.Proof. See [13, Lemma 5.2.6 p. 107. ].
Theorem 4.2.
The variety
T A α has SU P AP . roof. Since
T A α is defined by positive equations then it is canonical. In thiscase L = Cm − T A α consists of frames ( V , C i , S ji , S i,j ) such that if s ∈ V , then[ i, j ] | s ∈ V and [ i | j ] | s is in V . Moreover, ( x, y ) ∈ C i iff x and y agree off i ,( x, y ) ∈ S ji iff [ i | j ] | x = y and same for S ij . The first order correspondants ofthe positive equations translated to the class of frames will be Horn formulas,hence clausifiable and so L is closed under finite zigzag products. ACKNOWLEDGEMENT . We are grateful to Mikl´os Ferenczi, for hisfruitful discussion.
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