Cyclic congruences of slim semimodular lattices and non-finite axiomatizability of some finite structures
NNON-FINITE AXIOMATIZABILITY OF SOME FINITESTRUCTURES
G ´ABOR CZ´EDLI
Dedicated to B´ela Cs´ak´any on his ninetieth birthday (2022)
Abstract.
We prove that finite bipartite graphs cannot be axiomatized byfinitely many first-order sentences in the class of finite graphs. (The counter-part of this statement for all bipartite graphs in the class of all graphs is awell-known consequence of the compactness theorem.) Also, to exemplify thatour method is applicable in various fields of mathematics, we prove that nei-ther finite simple groups, nor the ordered sets of join-irreducible congruencesof slim semimodular lattices can be described by finitely many axioms in theclass of finite structures. Since a 2007 result of G. Gr¨atzer and E. Knapp,slim semimodular lattices have constituted the most intensively studied partof lattice theory and they have already led to results even in group theory andgeometry. Introduction
Finite model theory is a thriving part of mathematics. This is witnessed by,say, the monograph Libkin [19] with its 250 references or by the fact that, at thetime of writing, MathSciNet returns seven matches to the search “Title=(finitemodel theory) AND Publication Type=(Books)”. However, the following wordsof Fagin [11, page 4] from 1993 are still valid: “almost none of the key theoremsand tools of model theory, such as the completeness theorem and the compactnesstheorem, apply to finite structures”. This could be the reason that, as opposed to(classical unrestricted) model theory, finite model theory has not paid to much, ifany, attention to finite axiomatizability among finite structures.This paper deals with the axiomatizability of three different classes of finitestructures. We prove that none of these three classes can be defined by a finite setof first-order sentences within the class of finite structures. The first class consistsof all finite bipartite graphs . While it is a trivial consequence of the compactnesstheorem that the class of all (not necessarily finite) bipartite graphs is not finitelyaxiomatizable, our result on the finite case is a bit more involved. The secondclass consists of all finite simple groups . Although our results on these two classesare not surprising, their proofs exemplify that, in “lucky cases”, some methods of(classical) model theory are applicable even in the finite world on condition that
Date : January 31, 2021
Always check the author’s web page for possible updates.2020
Mathematics Subject Classification.
Key words and phrases.
Finite model theory, non-finite axiomatizability, finite axiomatizabil-ity, finite bipartite graphs, finite simple group, join-irreducible congruence, congruence lattice,slim semimodular lattice, finite propositional logic, first-order language.This research was supported by the National Research, Development and Innovation Fund ofHungary under funding scheme K 134851. a r X i v : . [ m a t h . L O ] J a n G. CZ´EDLI sufficiently many of the structures we deal with are not too complicated or powerfultheorems apply to them.In case of the third class, whose definition with an appropriate introduction ispostponed to Section 4, our result is the opposite of what has previously beenconjectured. Here we only note that Section 4, containing the main result of thepaper, belongs to the most intensively studied part of lattice theory.
Prerequisites.
The results of Sections 2 and 3 are easy to understand for allmathematicians and even their proofs are readable for those who have ever met theconcept of ultraproducts. Section 4 is intended for lattice theorists.2.
Non-finite axiomatizability of bipartite graphs
We begin this section with recalling some known concepts and facts; they willalso be needed in the subsequent sections. By a finite signature we mean a tuple σ = (cid:10) p, q, (cid:104) R , r (cid:105) , . . . (cid:104) R p , r p (cid:105) , (cid:104) F , f (cid:105) , . . . (cid:104) F q , f q (cid:105) (cid:11) (2.1)where p, q ∈ N = { , , , . . . } , R , . . . , R p are relation symbols , F , . . . , F q are function symbols , and these symbols are of arities r , . . . , r p , f , . . . , f q ∈ N ,respectively. A structure of type σ or, shortly, a σ -structure is a (1 + p + q )-tuple A = (cid:104) A, R A , . . . , R A p , F A , . . . , F A q (cid:105) or, shortly, (cid:104) A, R , . . . , R p , F , . . . , F q (cid:105) , where A , called the underlying set , is a nonempty set, R A i ⊆ A r i is a relation, and F A j : A f j → A is a map for all i ∈ { , . . . , p } and j ∈ { , . . . , q } . Structures will bedenoted by calligraphic capital letters A , B , . . . while their underlying sets withthe corresponding italic capitals A , B , . . . . In this paper,the first-order language with equality determinedby σ will be denoted by Lng( σ ). (2.2)In addition to the relation symbols and function symbols occurring in (2.1), Lng( σ )includes the equality symbol, which is always interpreted as the equality relation.To define the the (first-order) consequence relation modulo finiteness , denoted by | = fin , assume that Φ is a set of Lng( σ )-sentences and µ is an Lng( σ )-sentence. Then µ is a consequence of Φ modulo finiteness , in notation Φ | = fin µ, (2.3)if every finite σ -structure that satisfies all sentences belonging to Φ also satisfies µ .To see an example, we borrow the sentence λ k : ∃ x . . . ∃ x k (cid:94) ≤ i The class of finite bipartite graphs is not finitely axiomatizablemodulo finiteness. That is, there exists no finite set Σ of Lng( σ gr ) -sentences suchthat a finite graph A is bipartite if and only if each member of Σ holds in A . For comparison, we note the following folkloric fact. Remark 2.2. Let σ be a finite signature. If K is a class of finite σ -structures suchthat it is closed with respect to taking isomorphic copies, then there exists a setΦ of Lng( σ )-sentences such that a finite structure belongs to K if and only if itsatisfies every member of Φ. Proof of Remark 2.2. For each k ∈ N + , K contains finitely many k -element struc-tures (up to isomorphism). Hence there is an Lng( σ )-sentence ν k that holds exactlyin the k -element structures of K . Thus, we can let Φ := { λ k ⇒ ν k : k ∈ N + } . (cid:3) Proof of Proposition 2.1. For 2 ≤ n ∈ N + , the circle of length n is the graph C n with base set C n := { , , . . . , n − } and E C n = {(cid:104) x, y (cid:105) : | x − y | ∈ { , n − }} .Let J := { , , , , , , . . . } and J := { , , , , , , . . . } . (2.5)Note that C n is bipartite for all n ∈ J but C n is not bipartite if n ∈ J . The setof all subsets of J i will be denoted by P ( J i ). For i ∈ { , } , let U i be a nontrivialultrafilter over J i ; see, for example, Poizat [20] for this concept. What we need hereis that ∅ / ∈ U i ⊆ P ( J i ) and U i contains all cofinite subsets of J i ; a subset X ⊆ J i is cofinite if J i \ X is finite. Let A i = (cid:104) A i , E (cid:105) be the ultraproduct (cid:81) n ∈ J i C n /U i . Weknow from Frayne, Morel, and Scott [12] or from Keisler [17] thatan ultraproduct of finite structures modulo a nontrivial ultrafilteris either finite, or it has at least continuum many elements. (2.6)For i ∈ { , } , k ∈ N + , and λ k defined in (2.4), { n ∈ J i : λ k holds in C n } is acofinite set, whereby it belongs to U i . Hence, λ k holds in A i by Lo´s’s Theorem;see, for example, Theorem 4.3 in Poizat [20]. Since this is true for all k ∈ N + , A i is not finite. Also, it has at most continuum many elements since the cardinalityof the direct product (cid:81) n ∈ J i C n is continuum. Thus (2.6) gives that, for i ∈ { , } ,the cardinality of A i is continuum; in notation, | A i | = 2 ℵ . (2.7)Note that the subsequent sections will reference (2.7) in connection with otherstructures defined by similar ultraproducts of finite structures.The Z -chain is the graph C ∞ with the set C ∞ := Z of integer numbers as vertexset and E C ∞ := {(cid:104) x, y (cid:105) : | x − y | = 1 } . There is an Lng( σ gr )-sentence expressingthat for every element x there are exactly two elements y such that xEy . Apartfrom n = 2, this sentence holds in all C n . Hence, by Lo´s’s Theorem again, thissentence holds in A and A . This yields that A i is the disjoint union of copiesof Z -chains and circles C k , 3 ≤ k ∈ N . However, for each 3 ≤ k ∈ N + , there isan Lng( σ gr )-sentence expressing that C k is not a subgraph. This sentence holds in C n for all n belonging to the cofinite set J i \ { k } , whereby Lo´s’s Theorem givesthat this sentence also holds in A i . Hence, A i contains no circle. Consequently,for i ∈ { , } , there is a cardinal number κ i such that A i is the disjoint unionof κ i many copies of Z -chains. Combining | C ∞ | = ℵ with (2.7), it follows that κ = 2 ℵ = κ . Thus, A and A are isomorphic graphs; in notation, A ∼ = A . G. CZ´EDLI Next, for the sake of contradiction, suppose that Proposition 2.1 fails. Then,using that finitely many sentences can always be replaced by their conjunction,there exists a single sentence ϕ such that for every finite graph B , ϕ holds in B ifand only if B is a bipartite graph. In particular, ϕ holds in C n for all n ∈ J but it fails in C m for all m ∈ J . (2.8)By Lo´s’s Theorem, ϕ holds in A . Hence, by the isomorphism A ∼ = A , ϕ holds in A , too. Using Lo´s’s Theorem again, we obtain that the set { m ∈ J : ϕ holds in C m } belongs to the ultrafilter U . This contradicts the fact that this set is emptyby (2.8), completing the proof of Proposition 2.1. (cid:3) Groups Using the terminology of Proposition 2.1, we have the following statement. Proposition 3.1. The class of finite simple groups is not finitely axiomatizablemodulo finiteness.Proof. Since lots of arguments used in Proposition 2.1 apply here, we give lessdetails. According to (2.1), the signature σ gr for groups is chosen so that p = 0, q = 1, f = 2, and F is “+”. For the sake of contradiction, suppose that thereexists an Lng( σ gr )-sentence ϕ that holds in all finite simple groups but it fails in allfinite non-simple groups. For n ∈ N + , the cyclic group of order n will be denotedby C n . Let p < p < p < . . . be the list of all prime numbers, and define q j := p j p j +1 for j ∈ N + . Take a nontrivial ultrafilter U over N + . Let A and A be the ultraproducts (cid:81) n ∈ N + C p i /U and (cid:81) n ∈ N + C q i /U , respectively. Observe that(2.7) is still valid; see the sentence right after it. For k ∈ N + , define the followingsentence of Lng( σ gr ) with k occurrences of y : η k : ∀ x ∃ y (cid:0) ( . . . ( y + y ) + y ) + . . . ) + y = x ) . Basic facts about linear congruences yield that the sets { n ∈ N + : η k holds in C p n } and { n ∈ N + : η k holds in C q n } are cofinite and so they belong to U . Similarly,with k + 1 occurrences of x before the first equality sign, if we define τ k : ∀ x (cid:16)(cid:0) ( . . . ( x + x ) + x ) + . . . ) + x = x ⇒ ∀ y ( x + y = y ) (cid:17) , then both { n ∈ N + : τ k holds in C p n } and { n ∈ N + : τ k holds in C q n } are cofiniteand belong to U . Hence, by Lo´s’s Theorem, η k and τ k hold in A i for all k ∈ N + and i ∈ { , } . Therefore, the abelian groups A and A are torsion-free (by thesentences τ k ) and divisible (by the η k ). Consequently, they are direct sums of copiesof the additive group (cid:104) Q , + (cid:105) of rational numbers; see, for example, Kurosh [18, page165] or use the straightforward fact that a torsion-free and divisible abelian groupcan be considered a vector space over the field of rational numbers. (2.7) impliesthat each of A and A has 2 ℵ -many direct summands. Hence, A ∼ = A .For all n ∈ N + , C p n is a simple group and so it satisfies ϕ . Lo´s’s Theorem givesthat ϕ holds in A , whereby it holds in A since A ∼ = A . Using Lo´s’s Theoremagain, we obtain that the set I := { n ∈ N + : ϕ holds in C q n } belongs to the ultrafilter U . But none of the groups C q n is simple, so none of them satisfies ϕ , whence I = ∅ .The contradiction ∅ = I ∈ U completes the proof of Proposition 3.1. (cid:3) Note that (2.7) and the structure theorem of torsion-free divisible abelian groupsin the proof above were only used to conclude that A ∼ = A , but this isomorphism ON-FINITE AXIOMATIZABILITY OF SOME FINITE STRUCTURES 5 was only needed to ensure that A and A are elementarily equivalent. Thereis another way to ensure this elementary equivalence that relies neither on (2.7),nor on the above-mentioned structure theorem: one can use the description ofelementary equivalence of abelian groups given by Szmielew [21]. However, the useof this description would require further Lng( σ gr )-sentences and would make theproof more complicated.4. The ordered sets of join-irreducible congruences of slimsemimodular lattices Brief introduction to slim semimodular lattices. We assume that the readerhas some basic familiarity with lattices; if not then a few parts of Burris andSankappanvar [1] or Davey and Priestley [10] or Gr¨atzer [13] are recommended.A lattice L = (cid:104) L ; ∨ , ∧(cid:105) is semimodular if for any x, y, z ∈ L , the covering relation x ≺ y implies that x ∨ z ≺ y ∨ z or x ∨ z = y ∨ z . The lattice L is slim if it is finiteand the (partially) ordered set J ( L ) = (cid:104) J ( L ) , ≤(cid:105) of its join-irreducible elements isthe union of two chains. We know from Cz´edli and Schmidt [9, Lemma 2.3] that forfinite semimodular lattices, this definition of slimness is equivalent to the originalone, which is due to Gr¨atzer and Knapp [14] but not recalled here. We also knowfrom Cz´edli and Schmidt [9, Lemma 2.2] that slim lattices are planar ; however,the term “slim, planar, semimodular lattice” frequently occurs in the literaturesince the original concept of slimness did not imply planarity. Here we write “slimsemimodular lattices” and these lattices are automatically finite and planar. Asusual, the set of congruence relations of a lattice L form a lattice, the congruencelattice Con L of L . The study of congruence lattices of slim semimodular latticesbegan with Gr¨atzer and Knapp [15]. These congruence lattices Con L are distribu-tive. Hence, by the classical structure theorem of finite distributive lattices, seeGr¨atzer [13, Theorem II.1.9] for example, these congruence lattices are economi-cally described by simpler and smaller structures: the ordered sets J (Con L ) oftheir join-irreducible elementsSeveral properties of the ordered sets J (Con L ) determined by slim semimodularlattices L have been discovered; they are summarized in Cz´edli [5] and Cz´edli andGr¨atzer [7]. In fact, the attempt to characterize these J (Con L ) served as the mainmotive to deal with slim semimodular lattices. For surveys of these lattices, see thebook chapter Cz´edli and Gr¨atzer [6] and Section 2 of Cz´edli and Kurusa [8]. Here,as an appetizer to this section of the paper, we only mention that slim semimodularlattices were used to strengthen the Jordan–H¨older Theorem for groups from thenineteenth century, see Cz´edli and Schmidt [9] and Gr¨atzer and Nation [16], andthey have led to results in geometry, see Cz´edli [3]–[4] and Cz´edli and Kurusa [8]together with the survey given in it. Since 2007, when G. Gr¨atzer and E. Knapp[14]introduced slim semimodular lattices, the study of these lattices has been the mostintensive part of lattice theory. Indeed, at the time of writing, the MathSciNetsearch “Anywhere=(slim and semimodular)” returns 22 matches. The main result of the paper and its proof. In harmony with (2.1), we assumethat ordered sets are of type σ ord = (cid:10) , , (cid:104)≤ , (cid:105) (cid:11) . Using this notation and (2.2),we formulate our main result as follows. Theorem 4.1. The class of ordered sets of join-irreducible congruences of slimsemimodular lattices is not finitely axiomatizable modulo finiteness. That is, there G. CZ´EDLI exists no finite set Φ of Lng( σ ord ) -sentences such that a finite ordered set S = (cid:104) S, ≤(cid:105) is isomorphic to the ordered set J (Con L ) = (cid:104) J (Con L ) , ≤(cid:105) of some slimsemimodular lattice L if and only if all members of Φ hold in S .Proof. Suppose the contrary. Then, as in the proof of Proposition 2.1, we can picka single Lng( σ ord )-sentence ϕ such that a finite ordered set S satisfies ϕ if and onlyif S ∼ = J (Con L ) for a slim semimodular lattice L . For 2 ≤ n ∈ N + , the n -crown K n is the 2 n -element ordered set with maximal elements a , a , . . . , a n − and minimalelements b , b , . . . , b n − such that, for i, j ∈ { , , . . . , n − } , b i ≤ a j if and only if i = j or i + 1 ≡ j (mod n ). For n = 8, K n is drawn below.We let J = { , , , . . . } and J = { , , , . . . } . Take a nontrivial ultrafilter U i over J i . For i ∈ { , } , let A i be the ultraproduct (cid:81) n ∈ J i K i /U i . We know from(2.7) and the sentence following it that | A | = | A | = 2 ℵ . Although, to save space,we do not give all of them in details, we have the following Lng( σ ord )-formulas. α ( x ): ∀ y ( x ≤ y ⇒ y ≤ x ), which expresses that x is a maximal element. β ( x ): ∀ y ( y ≤ x ⇒ x ≤ y ), which expresses that x is a minimal element. δ : ∀ x , exactly one of α ( x ) and β ( x ) holds. δ : ∀ x , if α ( x ), then there are exactly two elements y such that β ( y ) and y ≤ x . δ : ∀ x , if β ( x ), then there are exactly two elements y such that α ( y ) and x ≤ y . ξ m : there are no elements forming a subset order isomorphic to K m .The ordered set F = (cid:104){ a j : j ∈ Z } ∪ { b j : j ∈ Z } , ≤(cid:105) such that α ( a j ), ¬ β ( a j ), β ( b j ),and ¬ α ( a j ) for all j ∈ Z and, in addition, b j ≤ a s if and only if s ∈ { j, j + 1 } willbe called an (infinite) fence ; see the Figure below.Recall that for ordered sets W h = (cid:104) W h , ≤ h (cid:105) , h ∈ H , we obtain the cardinal sum W = (cid:104) W, ≤(cid:105) of these ordered sets by letting W be the disjoint union of the W h , h ∈ H , and defining ≤ as the union of the ≤ h , h ∈ H . Since δ , δ , and δ hold in K n for all n ∈ J ∪ J and, for each m ≥ 2, so does ξ m for all n ∈ ( J ∪ J ) \ { m } ,Lo´s’s Theorem yields that δ , δ , δ , and, for all m ∈ N + \ { } , ξ m hold in A and A . Therefore, for i ∈ { , } , we conclude that each element of A i belongs to aunique fence and A i is the cardinal sum of some copies, say κ i copies, of fences.Using | A | = 2 ℵ = | A | , we obtain that κ = 2 ℵ = κ . Therefore, A ∼ = A .The rest of the proof relies heavily on Cz´edli [2] and mainly on [5]; these twopapers should be near. In particular, the notation and the concepts not definedhere are given there. By the “Bipartite maximal elements property”, see Corollary3.4 in [5], K n ∼ = J (Con L ) with a slim semimodular L cannot hold if n ∈ J . Hence, ϕ fails in K n for n ∈ J , and Lo´s’s Theorem gives that ϕ does not hold in A .Next, we assume that n ∈ J . Let k := n/ 2. To construct a lattice, we beginwith the direct square of the ( k + 1)-element chain; it is a distributive lattice calleda grid . For n = 8, this grid consists of the pentagon-shaped elements in Figure 1.Going downwards, we label the edges on the upper left boundary by a , a , . . . , Temporary note: see for their preprints. ON-FINITE AXIOMATIZABILITY OF SOME FINITE STRUCTURES 7 Figure 1. L a n − . Also, we label the edges on the upper right boundary by a , a , . . . , a n − ,going downwards again. In this way, we have labeled the non-vertical thick edgesof Figure 1. At this stage, the circle-shaped elements and the edges having (atleast one) circle-shaped endpoints are not present. The edges of the grid determine k many 4-cells (that is, squares) in the plane. We obtain a slim semimodularlattice L n from the grid in n steps in the following way. First, we insert a fork(that is, a multifork of rank 1) into RightEnl( a ) ∩ LeftEnl( a ); this intersection isthe uppermost grey-filled rectangle (which happens to be a square) in the figure.This insertion brings the b -labeled thick vertical edge in. (Since there would notbe enough room otherwise, the label of a vertical thick edge is always below theedge in Figure 1; note that the thick edges are exactly the labeled edges.) In thesecond step, we insert a fork into RightEnl( a ) ∩ LeftEnl( a n − ), understood in thelattice obtained in the previous step, of course. In the figure, this step brings the b -labeled thick vertical edge in, and the intersection in question as well as thesubsequent intersections are grey-filled. In the third step, we insert a fork intoLeftEnl( a ) ∩ RightEnl( a ) and we obtain the b -labeled thick vertical edge. Andso on, inserting a fork into RightEnl( a ) ∩ LeftEnl( a ), LeftEnl( a ) ∩ RightEnl( a ),RightEnl( a ) ∩ LeftEnl( a ), . . . , RightEnl( a n − ) ∩ LeftEnl( a n − ), one by one andin this order, we obtain the thick vertical edges with labels b , b , b , . . . , b n − ,respectively. After performing these steps, we obtain the required lattice L n . For n = 8, L n = L is given in Figure 1. By Theorem 3.7 of Cz´edli [2], L n is a slimsemimodular lattice. By (the Main) Lemma 2.11 of Cz´edli [5], K n ∼ = J (Con L n ).This isomorphism and the choice of ϕ gives that ϕ holds in K n . This is true forall n ∈ J , whereby Lo´s’s Theorem implies that ϕ holds in A . But this is acontradiction since A ∼ = A but we have previously seen that ϕ does not hold in A . The proof of Theorem 4.1 is complete. (cid:3) G. CZ´EDLI References [1] Burris, S. and Sankappanavar, H. 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