Bare canonicity of representable cylindric and polyadic algebras

We show that for finite n at least 3, every first-order axiomatisation of the varieties of representable n-dimensional cylindric algebras, diagonal-free cylindric algebras, polyadic algebras, and polyadic equality algebras contains an infinite number of non-canonical formulas. We also show that the class of structures for each of these varieties is non-elementary. The proofs employ algebras derived from random graphs.


Introduction
The notion of the canonical extension of a boolean algebra with operators (or 'BAO') was introduced by Jónsson and Tarski in a classical paper [14], generalising a construction of Stone [21].It is an algebra whose domain is the power set of the set of ultrafilters of the original BAO, and its operations are induced from those of the BAO in a natural way.Canonical extensions are nowadays a key tool in algebraic logic, with a multitude of uses and generalisations.
A class of BAOs is said to be canonical if it is closed under taking canonical extensions.In this paper we are concerned with the classes of representable n-dimensional cylindric algebras, diagonal-free cylindric algebras, polyadic algebras, and polyadic equality algebras, for finite n ≥ 3.These four classes are varieties.They are non-finitely axiomatisable, and many further 'negative' results on axiomatisations are known (e.g., [1,20]).However, the classes are canonical.Now [14] already established that positive equations are preserved by canonical extensions, and more generally, Sahlqvist equations are also preserved (see, e.g., [2]).This may suggest that the four classes might be axiomatisable by positive or Sahlqvist equations.
It turned out that the representable cylindric algebras are not Sahlqvist axiomatisable [22, footnote 1].In this paper, we extend this result to a wider class of axioms and to all four classes.A first-order sentence is said to be canonical if the class of its BAO models is canonical.Although some syntactic classes of canonical sentences (such as Sahlqvist equations) are known, canonicity is a semantic property that cannot be easily defined syntactically.For example, there is no algorithm to decide whether an equation is canonical [15,Theorem 9.6.1].The goal of this paper is to show that there is no canonical axiomatisation of any of the four classes listed above.In fact, we will show that any first-order axiomatisation of any of them contains infinitely many non-canonical sentences.We say that a canonical class of BAOs with this property is barely canonical.Although the class is canonical, its canonicity emerges only 'in the limit' and does not reside in any finite number of axioms for it, however they are phrased.
There are a few related results in the literature.The class of representable relation algebras, proved to be canonical by Monk (reported in [17]), was shown in [12] to be barely canonical.Our proof in the current paper is similar but somewhat simpler: the use of finite combinatorics (finite Ramsey theorem, etc) in [12] is replaced here by the use of first-order compactness.Bare canonicity of the 'McKinsey-Lemmon' modal logic was shown in [7].
We sketch the rough outline of the proof.Our aim is to convey the idea quickly, and the description will not be completely accurate in detail.Our construction uses polyadic-type algebras built from graphs.They are polyadic expansions of cylindrictype algebras constructed from graphs in [11], where it was shown (roughly) that such an algebra is representable if and only if its base graph has infinite chromatic number.(This was used in [11] to prove that the class of structures for the variety of representable n-dimensional cylindric algebras (finite n ≥ 3) is non-elementary, a result generalised to diagonal-free, polyadic, and polyadic equality algebras in Theorem 8.3 below.)Here, we will cast this work in a wider setting by defining an elementary class K of three-sorted structures comprising a polyadic equality-type algebra A, a graph G, and a boolean algebra B of subsets of G.We will show that representability of A is equivalent to G having infinite chromatic number in the sense of B. Both these properties can be defined by first-order theories, which therefore have the same models modulo the theory defining K.It follows by compactness that if the class of representable algebras had a first-order axiomatisation using only canonical sentences, there would be a function f : ω → ω such that whenever an algebra A has chromatic number at least f (k) (in the sense of some three-sorted structure), its canonical extension has chromatic number at least k.We then borrow from [12] an inverse system of finite (random) graphs of chromatic number m whose inverse limit has chromatic number k, for any chosen 2 ≤ k < m < ω.Using some results of Goldblatt [6] connecting canonical extensions with inverse limits, this yields an algebra of chromatic number m whose canonical extension has chromatic number k.Since k, m are arbitrary, no function f as above can exist.A slight extension of the argument, using a little more compactness, shows that any first-order axiomatisation of the representable algebras has infinitely many non-canonical sentences.

Layout of paper
In Section 2 we recall some basic notions of algebras of relations, representability, duality and canonicity.We define polyadic equality-type algebras over graphs in Section 3, and abstract generalisations of them in Section 4, where we also ascertain some of their elementary properties.This is continued in Section 5, where we study their ultrafilters.In Section 6 we introduce approximations to representations by means of systems of ultrafilters called 'ultrafilter networks', and lower-dimensional approximations of them called 'patch systems'.This will allow us to prove in Section 7 that (roughly) an abstract algebra is representable if and only if its associated graph has infinite chromatic number.Assuming an axiomatisation with only finitely many non-canonical formulas, we use direct and inverse systems in Section 8 to build an algebra that satisfies an arbitrary number of axioms, while its canonical extension satisfies only a bounded number, and thus obtain a contradiction.Section 9 lists some open problems.
Notation We use the following notational conventions.We usually identify (notationally) a structure, algebra, or graph with its domain.For signatures L ⊆ L and an L -structure M , we write M L for the L-reduct of M .
Throughout the paper, the dimension n is a fixed finite positive integer and n is at least 3.It will often be implicit that cylindric algebras etc. are n-dimensional and that i, j, k, m, etc., denote indices < n.We identify a non-negative integer m with the set {0, 1, . . ., m − 1}.If V is a set, we write [V ] m for the set of subsets of size m of V. We write ω for the first infinite ordinal number.℘(S) denotes the power set of a set S.
For a function f : X → Y we write dom f for its domain, im f for its image, and f [X ] for {f (x ) | x ∈ X } when X ⊆ X.We use similar notation for m-ary functions, for m < ω -e.g., in Definition 2.7.For functions f, g, we write f • g for their composition: f • g(x) = f (g(x)).We omit brackets in function applications when we believe it improves readability.By α U , where α is an ordinal, we denote the set of functions from α to U , so an α-ary relation on U is a subset of α U .To keep the syntax similar to the finite case, we write x i for x(i) if x ∈ α U and i < α.

Algebras of relations
In this paper, we will consider four types of algebra: cylindric-type algebras, diagonalfree cylindric-type algebras, polyadic-type algebras, and polyadic equality-type algebras, all of dimension n.They differ in their signatures and notion of representation.Here, we define them formally and recall some aspects of duality and canonicity for them.

Signatures and algebras
Definition 2.1.We let 1. L BA = {+, −, 0, 1} denote the signature of boolean algebras, Here, the c i ('cylindrifications') and s σ ('substitutions') are unary function symbols and the d ij ('diagonals') are constants.By a cylindric-type algebra, we mean simply an algebra of signature L CAn .Diagonal-free cylindric-type algebras, polyadic-type algebras, and polyadic equality-type algebras are defined analogously for the other signatures.
Our concern in this paper is with representable algebras of these four kinds, but we briefly note that abstract algebras have been defined as well: namely, cylindric algebras, diagonal-free cylindric algebras, polyadic algebras, and polyadic equality algebras.They are algebras of the above types that satisfy in each case a finite set of equations that can be found in [9,10].In particular, cylindric algebras are defined in [9, Definition 1.1.1].We will not use the formal definition so we do not recall it here, but the proofs of some later lemmas will be easier for readers familiar with basic computations in cylindric algebras.The material in [9, §1] is easily enough for what we need.Readers not so familiar can easily verify our claims directly in the specific algebras we are working with.

Representations
Natural examples of each kind of algebra arise from algebras of n-ary relations on a set.Definition 2.2.A polyadic equality set algebra is a polyadic equality-type algebra of the form where U is a non-empty set, V = n U , and A polyadic set algebra (cylindric set algebra) is the reduct of a polyadic equality set algebra to the signature L PAn (respectively, L CAn ).Since L Dfn has no operations connecting two different dimensions, a diagonal-free cylindric set algebra is defined rather differently, as an L Dfn -algebra of the form where U 0 , . . ., U n−1 = ∅, V = i<n U i , and 3. An L PEAn -algebra is said to be representable if it is isomorphic to a subalgebra of a product of polyadic equality set algebras.The isomorphism is then called a representation.The class of all representable polyadic equality algebras of dimension n is called RPEA n .
Exactly analogous definitions are made for L Dfn , L CAn , and L PAn , using the appropriate set algebras in each case.The classes of representable algebras for these are, respectively, RDf n , RCA n , and RPA n .

Atom structures
We now recall a little duality theory, leading to canonicity, the topic of the paper.For more details, see, e.g., [14,2] and [9, §2.7].
Definition 2.4.Let L ⊇ L BA be a functional signature (i.e., one with only function symbols and constants).We write L + for the relational signature consisting of an (n + 1)-ary relation symbol R f for each n-ary function symbol f ∈ L \ L BA .By an (L)-atom structure, we will simply mean an L + -structure.We will sometimes refer to the elements of an atom structure as atoms.
Given an L-atom structure S = (S, R f | R f ∈ L + ), we write S + for its complex algebra: where each f ∈ L BA is interpreted in the natural way as a boolean operation on ℘(S), and and X 1 , . . ., X n ⊆ S. We identify each s ∈ S with the atom {s} of S + .
For the particular signature L PEAn , we will be defining atom structures in which R ci is an equivalence relation and R −1 sσ a function.So we adopt a slightly different definition of atom structure that is a little easier to specify in practice.Definition 2.5.A polyadic equality atom structure is a structure where D ij ⊆ S, ≡ i is an equivalence relation on S, and − σ : S → S is a function.We regard S as a standard L PEAn -atom structure in the sense of Definition 2.4 by interpreting R dij as D ij , R ci as ≡ i , and letting R sσ (s, t) iff t σ = s.

Canonicity
One source of atom structures is from boolean algebra with operators (BAOs).These originated in [14], where they were called 'normal BAOs', and they are now familiar: see, e.g., [2].Let L be a functional signature containing L BA .Definition 2.6.An L-BAO is an L-structure whose L BA -reduct is a boolean algebra and in which each f ∈ L \ L BA defines a function that is normal (its value is zero whenever any argument is zero) and additive in each argument.
For example, if S is an L-atom structure then S + is an L-BAO (note that constants are vacuously normal and additive).Any algebra in RDf n , RCA n , RPA n , and RPEA n is easily checked to be a BAO for its signature.Definition 2.7.Let B be an L-BAO.We define the ultrafilter structure B + to be the L-atom structure which has the set of ultrafilters (of the boolean reduct) of B as domain and, for any n-ary function symbol f ∈ L \ L BA and µ 0 , . . ., µ n−1 , ν ∈ B + , Canonical extensions were introduced in [14], where it was shown that there is a canonical embedding of B into B σ given by b → {ν ∈ B + | b ∈ ν}, so justifying the use of 'extension'.Canonical extensions of cylindric algebras are studied in [9, §2.7].Canonical varieties in general have been intensely studied, for example by Goldblatt [5], and it is not hard to derive the following well known result.The proof we give follows [5]: [5,Theorem 4.6] proves by a stronger version of the same method that RCA α and ICrs α are canonical varieties for every ordinal α.Canonicity of RCA n is proved in a different way in [9, p.459 , where U 0 , . . ., U n−1 = ∅ and R ci ((u 0 , . . ., u n−1 ), (v 0 , . . ., v n−1 )) iff u j = v j for each j ∈ n \ {i}.Let K PEA be the class of (L PEAn ) + -structures of the form , where U = ∅, R ci is defined in the same way as above, R dij ((u 0 , . . ., u n−1 )) iff u i = u j , and R sσ ((u 0 , . . ., u n−1 ), (v 0 , . . ., v n−1 )) iff u i = v σ(i) for each i < n.Let K PA , K CA be the class of reducts of structures in K P EA to the signatures (L PAn ) + and (L CAn ) + , respectively.
We now assume familiarity with the notation of [5].By Theorem 4.5 of [5], if K is a class of atom structures with PuK ⊆ HSUdK, then SCmSUdK is a canonical variety.By Theorem 2.2(2,5) of [5], PCm = CmUd and SUd = UdS, so SCmSUdK = SPCmSK.Now let K ∈ {K Df , K PEA , K PA , K CA }.Then K is closed under ultraproducts, and under inner substructures (since no structure of the above forms has any proper inner substructures), so PuK ⊆ K = SK.Consequently, SPCmK -the closure of {S + | S ∈ K} under subalgebras of products -is a canonical variety.But it follows from the definitions that SPCmK PEA = RPEA n , and similar results hold for the other three classes.
Notwithstanding this proposition, we will show that any first-order axiomatisation of any of these four varieties requires infinitely many non-canonical sentences.

Algebras from graphs
Here we will describe how to obtain polyadic equality type algebras from graphs.In this paper, graphs are undirected and loop-free.Recall that a set of nodes of a graph is independent if there is no edge between any two nodes in the set.

Atom structures from graphs
The first step is given by the following definitions (adapted from [11,Definition 3.5]), which construct a polyadic equality atom structure from a graph.
Notation.We let Eq(n) denote the set of equivalence relations on n.If ∼ ∈ Eq(n) and i < n, we will write ∼ i for the restriction of ∼ to n \ {i}.Definition 3.1.Let Γ = (V, E) be a graph.We let Γ × n denote the graph consisting of n copies of Γ with all possible additional edges between copies.Definition 3.2.Fix a graph Γ.Let S(Γ) be the set of all pairs (K, ∼), where K : n → Γ × n is a partial map and ∼ an equivalence relation on n that satisfies the following: 1.If |n/∼| = n, then dom(K) = n and im(K) is not independent.
For (K, ∼), (K , ∼ ) ∈ S(Γ) and i, j < n, we will write K(i) = K (j) if either K(i) and K (j) are both undefined, or they are both defined and are equal.According to this, if i ∼ j then K(i) = K(j).
Definition 3.4.Let Γ be a graph.The polyadic equality atom structure is defined as follows: 2. ≡ i is the equivalence relation on S(Γ) given by: (K, ∼) ≡ i (K , ∼ ) if and only if K(i) = K (i) and ∼ i = ∼ i for i < n.
3. For each σ : n → n, the map − σ : S(Γ) → S(Γ) is given by: (K, ∼) σ = (K σ , ∼ σ ), where We leave it to the reader to check that (K σ , ∼ σ ) is well defined and in S(Γ), and that K σ is determined by K and σ even though we cannot in general recover ∼ σ from them.Note that if σ is one-one then K σ = K • σ.Definition 3.5.We write A(Γ) for the n-dimensional polyadic equality type algebra At(Γ) + .Explicitly, where d ij = D ij as above, and for X ⊆ S(Γ), A(Γ) is the expansion to the signature of polyadic equality algebras of a cylindrictype algebra, also written A(Γ), that was defined in [11].So some results proved for it also apply to the A(Γ) defined above.Here is one (another is in Proposition 4.11 below): Proposition 3.6.Let Γ be a graph.Then the cylindric reduct of A(Γ) is an n-dimensional cylindric algebra.
4 Algebra-graph systems Proposition 3.6 establishes a relation between graphs and cylindric algebras.However, we need to study this relationship in a more abstract setting.We need to pick out certain elements, so that all the elements beneath are i-distinguishing and thus have K(i) defined on them.Definition 4.2.Let A be a cylindric-type or polyadic equality-type algebra.For i < n, define

Definitions
We generally take F i to be an element of the algebra under consideration (here, A), though sometimes we regard it as an L AGS -term.
Remark.Clearly, for an algebra from a graph A(Γ), F i is just the sum of all the idistinguishing atoms.For with operations defined as follows: • The A-sorted and B-sorted symbols are interpreted on A(Γ), ℘(Γ × n) in the natural way.
• E is interpreted as the edge relation on Γ × n.
• We have H(x, y) if and only if there is < n such that x, y ∈ Γ × { }.
• The relation ∈ denotes membership of elements of Γ × n in the sets that are elements of ℘(Γ × n).
• Finally, we have We now define a theory that helps us talk about the subclass of all the L AGSstructures similar to the ones derived from graphs.
where ϕ is an L AGS -formula with no quantifiers over the A-sort.
We define U to be the set of A-universal sentences that are true in all L AGSstructures M (Γ) for graphs Γ.An L AGS -structure M that is a model of U is called an algebra-graph system.This definition ensures that a good number of first-order statements that hold for algebras from graphs, also hold in any algebra-graph system.It will allow us to prove many first-order statements for algebra-graph systems, by just showing they are Auniversal and hold in M (Γ) for every graph Γ.We will refer to this approach as the generalisation technique.

Basic properties of algebra-graph systems
Lemma 4.5.In any algebra-graph system M = (A, G, B), the cylindric reduct of A is a cylindric algebra, and B is a boolean algebra isomorphic to a subalgebra of ℘(G).
Proof.The first statement follows by the generalisation technique, as we know from Proposition 3.6 that an arbitrary algebra from a graph will satisfy all the axioms for cylindric algebras.These axioms are equations and can be recast in the obvious way as A-universal L AGS -sentences.A similar argument shows that B is a boolean algebra.Since the A-universal sentences are in U and so are true in M , the function So in any algebra-graph system (A, G, B), Lemma 4.5 allows us to regard B as a boolean algebra of subsets of G, and the L AGS -relation symbol '∈' as denoting genuine set membership.
Recall that F i = j<k<n, j,k =i −d jk from Definition 4.2.
Lemma 4.6.Let M = (A, G, B) be an algebra-graph system and i, j < n.Then Proof.It is enough to prove the lemma for algebras from graphs, as it is clearly a set of A-universal first-order sentences.But this is easy and was done in [11,Lemma 4.2].It is also easily seen to hold in cylindric algebras, of which A is one (by Lemma 4.5).
Now we examine the functions R i , S i .
Lemma 4.7.Let M = (A, G, B) be an algebra-graph system and let i, j < n be distinct.Then: Proof.It is again sufficient to show that this is true for any structure M (Γ) from Definition 4.3.Parts (i) and (ii) are easy and left to the reader.(iii) Let (K, ∼) ∈ a ≤ F i be arbitrary.Recall that , and hence This shows that a ≤ S i (R i (a)).
(iv) First, observe that For, we may define ≈ ∈ Eq(n) to be the (unique) i-distinguishing relation with i ≈ j and define K p by Then (K p , ≈) is certainly a valid element of At(Γ) contained in F i and d ij and with K p (i) = p, and it is clearly the only such atom.Returning to the lemma, it is clear that f (0) = 0 and f (a For the converse, let p ∈ B be given.By ( ‡) above, (K p , ≈) ∈ S i (B), and , and the converse is trivial.
Next, we examine the substitution operators.
(vi) If i / ∈ im σ then c i s σ a = s σ a, and if σ is one-one then c σ(i) s σ a = s σ c i a.
Proof.Again, it is enough to show that the lemma is true for an arbitrary algebra-graph system M (Γ) from a graph Γ, as all statements are definable by A-universal first-order sentences.
(i) By the definitions, s σ ∅ = ∅, s σ 1 = 1, and for any a, b ∈ A(Γ), (iv) By (i) and (iii), The last expression comprises some of the conjuncts (all of them, if This proves that c i s σ a ≤ s σ a.The converse is immediate by Lemma 4.5.Now suppose that σ : n → n is one-one.Then plainly, for any atoms (K, ∼), (K, ∼ ), we have

Simple algebras
Recall that a cylindric algebra A is simple if |A| > 1 and for any algebra A with cylindric signature, any homomorphism ϕ : A → A is either trivial or injective.We will see that the cylindric reduct of the algebra part of an algebra-graph system is simple, so that if it is representable, it has a representation that is just an embedding into a single cylindric set algebra.We deduce the following in a standard way.
Corollary 4.12.In every algebra-graph system (A, G, B), the cylindric-type reduct A L CAn of A is simple, as is each of its subalgebras.Thus ϕ is trivial if it is not injective.
Lemma 4.13.Let A ∈ RCA n be a representable cylindric algebra.If A is simple, then it has a representation that is an embedding into a single cylindric set algebra.
Proof.There is a representation h : A → k∈K S k , where K is an index set and for each k ∈ K, S k is a non-empty base set and Because h is injective and |A| > 1, the index set K = ∅.So choose ∈ K and let π be the projection of k∈K S k onto S .Then π • h is certainly a homomorphism and because injective and thus a representation that is an embedding into a single cylindric set algebra.

Ultrafilters
We now examine ultrafilters in algebra-graph systems.

Ultrafilter structures from algebra-graph systems
By the generalisation technique, if M = (A, G, B) is an algebra-graph system then A is an L PEAn -BAO, so its ultrafilter structure We view A + as a polyadic equality atom structure (Definition 2.5) by defining Lemma 5.1, the comment following it, and Lemma 5. 3(v) show that this gives a well defined polyadic equality atom structure which, when regarded as an L PEAn -atom structure as in Definition 2.5, yields the ultrafilter structure A + as above.
Lemma 5.1.Let M = (A, G, B) be an algebra-graph system and σ, τ : n → n.Then for any ultrafilter ν of A, the set ν σ is also an ultrafilter of A, and Proof.By Lemma 4.9, s σ : A → A is a boolean homomorphism.It is well known and easily seen that for boolean algebras B 1 , B 2 , the preimage of an ultrafilter of B 2 under a boolean homomorphism f : B 1 → B 2 is an ultrafilter of B 1 .So ν σ is an ultrafilter of A. By Lemma 4.9(ii), It follows that R sσ (µ, ν) iff µ ⊆ ν σ , iff ν σ = µ since both are ultrafilters.

Projections of ultrafilters
Definition 5.2.Let M = (A, G, B) be an algebra-graph system, let µ be an ultrafilter of A, and let i < n.We write µ(i) for the set Clearly, µ is i-distinguishing iff it does not contain any of the d jk for distinct j, k ∈ n \ {i}.In this case, µ(i) turns out to be an ultrafilter of B. The following lemma establishes this and other facts about projections of ultrafilters.
Lemma 5.3.Let M = (A, G, B) be an algebra-graph system, let i < n, and let µ, ν be ultrafilters of A.
(i) The projection µ(i) is an ultrafilter of B if µ is i-distinguishing, and B (that is, the improper filter on B), otherwise.
(iii) If i = j < n and β is an ultrafilter of B, then Let A i be the relativisation of the boolean reduct of A to F i .It is easily seen by the generalisation technique that . By (i), µ(j) is always a filter, so R i (a) ∈ µ(j).Thus µ(i) ⊆ µ(j).The converse inclusion holds by symmetry, so µ(i) = µ(j).
(iii) By Lemma 4.7(iv), the map a → R i (a • d ij ) is a boolean homomorphism from A to B. As α is the preimage of β under this map, it is an ultrafilter of A. The lemma also shows that R Plainly, α(i) ⊆ β, so as β is an ultrafilter of B, by (i) we have α(i) = β.
Let α be any ultrafilter of A with ).As µ and ν are ultrafilters, this proves (a).Hence also, We prove (b).If −F i ∈ µ, part (i) gives µ(i) = B = ν(i), proving (b).Assume then that F i ∈ µ.Then µ(i) and ν(i) are ultrafilters by part (i), so it is enough to show µ(i) ⊆ ν(i).Let B ∈ µ(i) be arbitrary.Take a ∈ µ such that B = R i (a).By assumption, c i a ∈ ν.Note that the following holds for all algebras from graphs: ( ⇐= ) For the converse, assume the hypotheses and let so D ∈ µ ∩ ν by (a).Now the following holds in algebras from graphs: , so we may pick (K , ∼ ) ∈ a with K (i) = K(i).If K(i) is undefined, let (K , ∼ ) ∈ a be arbitrary (we use a > 0 here).Since (K, ∼), (K , ∼ ) ∈ D, we have ∼ i = ∼ i , hence in the second case K (i) is also undefined and By the generalisation technique, the statement holds for (vi) Let B ∈ µ σ (i).Then B = R i (a) for some a ∈ µ σ , so s σ a ∈ µ and R j (s σ a) ∈ µ(j).By Lemma 4.9(v), which applies since σ[n \ {i}] = n \ {j}, we have R j (s σ a) ≤ R i (a) = B.As µ(j) is a filter, B ∈ µ(j) as well.As B was arbitrary, µ σ (i) ⊆ µ(j).
So by part (i), it only remains to show that if µ σ (i) is an ultrafilter of B then so is µ(j).But as σ[n \ {i}] = n \ {j}, the definition of F i and Lemma 4.9(i)(iii) yield

Networks and patch systems
In this section we introduce approximations to representations, called ultrafilter networks.They will be part of the game to construct representations.We will approximate the networks themselves by lower-dimensional objects that we call patch systems.
6.1 Ultrafilter networks Definition 6.1.Let X be a set, i < n, and v ∈ n X.
1.For w ∈ n X, we say v ≡ i w if v j = w j for all j < n, j = i.

If v
Definition 6.2.Let M = (A, G, B) be an algebra-graph system.A cylindric ultrafilter network over A is a pair N = (N 1 , N 2 ), where N 1 is a set and N 2 : n N 1 → A + is a map that satisfies the following for any v, w ∈ n N 1 : 1.For i, j < n, we have N is said to be a polyadic ultrafilter network if in addition: , we write k<ω N k for the ultrafilter network 2 ) (here we view the maps N k 2 formally as sets of ordered pairs).We will often write N for both N 1 and N 2 .

Patch systems
Patch systems provide a way to assign ultrafilters on a graph to (n − 1)-sized subsets, or 'patches', of a set of nodes.Definition 6.3.Let M = (A, G, B) be an algebra-graph system.A patch system for B is a pair P = (P 1 , P 2 ), where P 1 is a set and P 2 : [P 1 ] n−1 → B + assigns an ultrafilter of B to each subset of P 1 of size n − 1.
n is said to be P-coherent if the following is satisfied: For any B i ∈ P 2 (V \ {v i }) (i < n), there are p i ∈ G with p i ∈ B i for each i < n, such that {p 0 , . . ., p n−1 } is not an independent subset of G.The patch system P is said to be coherent if every set V ⊆ P 1 of size n is P-coherent.Lemma 6.4.Let M = (A, G, B) be an algebra-graph system and P = (P 1 , P 2 ) a patch system for B. Let V = {v 0 , . . ., v n−1 } ∈ [P 1 ] n and for each i < n, let V i = V \ {v i }.Then V is P-coherent if and only if there exists an ultrafilter µ of A that is i-distinguishing and with µ(i) = P 2 (V i ) for each i < n.
Proof. ( =⇒ ) Assume V is P-coherent.Define To show that µ 0 has the finite intersection property, it is sufficient to consider arbitrary B i ∈ P 2 (V i ) and prove that S 0 (B 0 ) • S 1 (B 1 ) • • • S n−1 (B n−1 ) = 0.By the P-coherence of V , we can find p i ∈ B i for each i < n such that {p 0 , . . ., p n−1 } is not an independent set.Now the following A-universal sentence holds in structures M (Γ), because there is an atom (K, ∼) that is i-distinguishing and such that K(i) = p i , for each i < n: We showed that the left hand side of the implication is satisfied, so the right hand side gives us that µ 0 has the finite intersection property.By the boolean prime ideal theorem, µ 0 extends to an ultrafilter µ of A. Since plainly Lemma 5.3(i), since both sides are ultrafilters of B.
( ⇐= ) Assume µ is an ultrafilter of A that is i-distinguishing for all i < n and with µ(i) = P 2 (V i ) for each i < n.Choose arbitrary Now the following A-universal sentence holds by definition in algebras from graphs, because we can take (K, ∼) ∈ x, and then im K is not independent and K(i) ∈ R i (x) for each i: So we can choose p 0 , . . ., p n−1 with p i ∈ R i (b) ⊆ R i (b i ) = B i and such that {p 0 , . . ., p n−1 } is not independent.We conclude that V is P-coherent.

Patch systems from cylindric networks
Here we show how to construct a coherent patch system from a cylindric ultrafilter network.We will need the following lemma to show that it is well defined.We adopt the standard notation that if i, j < n then [i/j] : n → n denotes the function given by [i/j](i) = j and [i/j](k) = k for k = i.Lemma 6.5.Let M = (A, G, B) be an algebra-graph system and N = (N 1 , N 2 ) a cylindric ultrafilter network over A. Let i, j < n and v, w ∈ n N 1 .Then: Proof.(i) We have that N 2 (v) F i if and only if it does not contain d jk for j < k < n and j, k = i.But this is true if and only if v is i-distinguishing by the definition of cylindric ultrafilter networks.
(ii) This is trivial if i = j, so suppose not.If k, l = j and (v . Since the indices cannot be i, this implies that [i/j](k) = [i/j](l).As k, l = j, this implies that k = l as required.
(iii) Write w = v • [i/j].Then w ≡ i v and w i = v j = w j .By the definition of ultrafilter network we have N 2 (v) ≡ i N 2 (w) and d ij ∈ N 2 (w).So by Lemma 5.3(iv) we have N 2 (v)(i) = N 2 (w)(i), and by (ii) of the same lemma, N 2 (w)(i) = N 2 (w)(j).
(iv) Assume the hypothesis.Now and similarly for w.So if v is not i-distinguishing then neither is w j-distinguishing, and by part (i) and Lemma 5.
So assume that v is i-distinguishing, and hence that w is j-distinguishing.We may suppose without loss of generality that i = j = 0 (by (ii,iii), we can just replace v by v • [i/0] and w by w • [j/0]).
The proof is by induction on the highest number v, w disagree on: So, using the induction hypothesis, we get The third part in the above lemma says that the ith projection is independent from the ith coordinate and the order of the elements in the vector.This allows us to define the following: Definition 6.6.Let M = (A, G, B) be an algebra-graph system and N = (N 1 , N 2 ) a cylindric ultrafilter network over A. We define ∂N to be the patch system (N 1 , P 2 ), where for each i < n and i-distinguishing v ∈ n N 1 .Proposition 6.7.Let M = (A, G, B) be an algebra-graph system and N = (N 1 , N 2 ) a cylindric ultrafilter network over A. Then ∂N is a well defined and coherent patch system for B.

Polyadic networks from patch systems
A patch system contains a lot of the information in an ultrafilter network.Here we show that given a coherent patch system P = (P 1 , P 2 ), we can always find ultrafilters to assign to n-tuples of P 1 respecting P 2 , and under fairly minimal conditions, they form a polyadic ultrafilter network.Lemma 6.8.Let M = (A, G, B) be an algebra-graph system and P = (P 1 , P 2 ) a coherent patch system for B. Let v ∈ n P 1 .Then there is an ultrafilter µ of A such that 1.For i, j < n, we have Proof.There are three cases.
(a) If |im(v)| = n, then by Lemma 6.4 there is an ultrafilter µ of A that is i-distinguishing and with µ(i) = P 2 ({v j | i = j < n}), for all i < n.
is an ultrafilter of A with F i , d ij ∈ µ (and hence F j ∈ µ by Lemma 4.6), so for each k, l < n we have By the generalisation technique, D is an atom of A (in an algebra from a graph it would just be (∅, ∼) where i ∼ j if and only if v i = v j ).We define µ to be the principal ultrafilter of A generated by D. Condition 2 holds vacuously as v is never i-distinguishing.Lemma 6.9.Let M = (A, G, B) be an algebra-graph system and P = (N 1 , P ) a coherent patch system for B. Suppose N 2 : n N 1 → A + is a function satisfying the following, for any v ∈ n N 1 : 1.For i, j < n, we have Proof.We check the conditions from Definition 6.2 defining ultrafilter networks.The first condition, that For the second condition, take i < n and v, w ∈ n N 1 with v ≡ i w.We require N 2 (v) ≡ i N 2 (w).By assumption (2) of the lemma, if v, w are i-distinguishing we have and if they are not, then by Lemma 5.3(i) we have Lastly we check the third condition for ultrafilter networks.Let σ : n → n, let w = v • σ, and let ∼ ∈ Eq(n) be given by i ∼ j iff v i = v j .Observe that i ∼ σ j iff w i = w j .We check that There are three cases.If |n/∼ σ | = n, then v • σ is one-one and the result is given.Suppose that |n/∼ σ | = n − 1.Let {i, j} be the unique ∼ σ -class of size 2. By condition 1 of the lemma, Now if k, l ∈ n \ {i} and σ(k) = σ(l), then certainly k ∼ σ l, so k = l by assumption on ∼ σ .Hence, σ is one-one on n \ {i}, so σ[n \ {i}] = n \ {l} for some l < n.We now obtain by Lemma 5.3(vi).
Finally suppose that |n/∼ σ | < n − 1.Then d ∼ σ is an atom of A -this is true in algebras from graphs, because we have d ∼ σ = {(∅, ∼ σ )}, so it holds for M by the generalisation technique.So N 2 (w) is the principal ultrafilter generated by d ∼ σ .
Let a ∈ N 2 (v) σ be arbitrary, so that s σ a ∈ N 2 (v).By the first part, d ∼ ∈ N 2 (v), so s σ a • d ∼ > 0. By Lemma 4.9(i) and (iv), s σ (a 0 as well.As d ∼ σ is an atom, we obtain a ≥ d ∼ σ and a ∈ N 2 (w).This shows that N 2 (v) σ ⊆ N 2 (w), and equality follows since by Lemma 5.1 both sides are ultrafilters of A.

Chromatic number and representability
Here we show that the chromatic number of a graph Γ and the representability of A(Γ) and its reducts are tied together.
Recall that the chromatic number of a graph is the size of the smallest partition into independent sets, or ∞ if no such partition exists.Although the chromatic number is in general not first-order definable, we can define an analogue for algebra-graph systems with the following formula.Definition 7.1.For each k < ω, we define the following L AGS -sentence: Then M = (A, G, B) |= θ k iff the chromatic number of G is larger than k 'as far as B can tell'.The true chromatic number of G may be smaller, but B contains no independent sets witnessing this.However, B's estimate is correct when B = ℘(G), as in structures of the form M (Γ).
Remark.If M = (A, G, B) is an algebra-graph system, we will say an element B ∈ B is an independent set, if there are no p, q ∈ B such that E(p, q).

Representable implies infinite chromatic number
This direction can be proved without further help, apart from some of the machinery from the preceding section and Ramsey's theorem.Proposition 7.2.Let M = (A, G, B) be an algebra-graph system in which B is infinite.If the diagonal-free reduct of A is representable, then M |= Θ.
Proof.Suppose for a contradiction that the reduct of A to the signature of diagonal-free cylindric algebras is representable but M |= θ k for some k < ω.
Recall (e.g., from [9, §1.6]) that for a ∈ A, ∆a = {i < n | c i a = a}.Define D = {a ∈ A | ∆a = n}, and let A be the closure of D under the boolean operations.We first claim that A is a subalgebra of A. By Lemma 4.5, the cylindric reduct of A is a cylindric algebra.By basic cylindric algebra, or the generalisation technique, ∆0 = ∆1 = ∆d ii = ∅ for i < n; also, ∆d ij = {i, j} for distinct i, j < n, so since n ≥ 3, ∆d ij = n; finally, if a ∈ A and i < n then i / ∈ ∆c i a.So all these elements are in D and hence in A .Obviously, A is closed under + and −.By Lemma 4.9(vi), D is closed under each s σ , so by Lemma 4.9(i), so is A .This proves the claim.Now let N = (A , G, B).We claim next that N is a substructure of M .Inspecting the function symbols of L AGS , it suffices to show that S i (B) ∈ A for every B ∈ B and i < n.But by Lemma 4.7(vi), i / ∈ ∆S i (B), so S i (B) ∈ D ⊆ A .This proves the claim.
As M |= U and all sentences in U are A-universal, it follows that N |= U.So N is also an algebra-graph system in which B is infinite.By Lemma 4.5 and Corollary 4.12, the cylindric reduct A L CAn is a simple cylindric algebra.It is generated by D, and its diagonal-free reduct is representable (since the diagonal-free reduct of A is).It follows from a theorem of Johnson [13,Theorem 1.8(i)] that A L CAn is representable as a cylindric algebra.So by Lemma 4.13, there is a cylindric representation h that embeds A L CAn into a single cylindric set algebra S = (℘( n S), ∪, \, ∅, n S, D S ij , C S i ) i,j<n with base set S.
Let N be the ultrafilter network with nodes S and N (s) = {a ∈ A | s ∈ h(a)} ∈ A + , for s ∈ n S.This is easily seen to be a well-defined cylindric ultrafilter network over A .Furthermore, by Proposition 6.7 we can make it into a well-defined and coherent patch system ∂N .Now M |= θ k means that the following is true in M and therefore N : So G is the union of k independent sets from B: say, B 0 , . . ., B k−1 .Since B is infinite, by Lemma 4.7(v) A is also infinite.As h is injective, S is infinite and therefore S as well.So we can choose infinitely many pairwise distinct elements s 0 , s 1 , . . .from S. Now define a map f : [ω] n−1 → k by letting f ({i 1 , . . ., i n−1 }) be the least j < k such that B j ∈ ∂N ({s i1 , . . ., s in−1 }).By Ramsey's theorem [19], we can choose the elements so that f has constant value c, say.Now consider {s 0 , . . ., s n−1 }.Since f is constant, B c ∈ ∂N ({s j | i = j < n}) for all i < n.Because ∂N is coherent, we can choose p 0 , . . ., p n−1 ∈ B c so that {p 0 , . . ., p n−1 } is not an independent set.But this is impossible since B c is independent.

Infinite chromatic number implies representable
For the other direction, we define a game that allows us to build a polyadic representation for A if M = (A, G, B) |= Θ (i.e., G has infinite chromatic number in the sense of B).Definition 7.3.Let M = (A, G, B) be an algebra-graph system.A game G(A) is an infinite sequence of polyadic ultrafilter networks built by the following rules.There are two players, named ∀ and ∃.The game begins with the (unique) one-point network N 0 .There are ω rounds.In round t < ω, the current network (at the start of the round) is N t and player ∀ chooses an n-tuple v ∈ n N t , a number i < n and an element a ∈ A such that c i a ∈ N t (v).The other player ∃ then has to respond with an ultrafilter network N t+1 ⊇ N t such that there is w ∈ n N t+1 with w ≡ i v and a ∈ N t+1 (w).She wins the game if she can play a network that satisfies these constraints in each round.Lemma 7.4.Let M = (A, G, B) be an algebra-graph system.If ∃ has a winning strategy in the game G(A), then A is a representable polyadic equality algebra.
Proof.By the downward Löwenheim-Skolem-Tarski theorem (see e.g.[3]), there is a countable elementary subalgebra A 0 of A. Let N 0 ⊆ N 1 ⊆ • • • be a play of the game G(A) in which ∀ plays every possible move in A 0 and ∃ uses her winning strategy in G(A) to respond.Define N = t<ω N t .This is certainly a polyadic ultrafilter network over A, as all the N t are polyadic ultrafilter networks.Now define: It can be checked that h is a homomorphism.Recall from Corollary 4.12 that A 0 is simple.So, since h(1) = n N = ∅ = h(0), the map h is injective.This shows that A 0 is representable, and because RPEA n is a variety, A is representable as well.
Remark.The converse of the lemma also holds, but is not needed here.
By the generalisation technique, in any algebra-graph system (A, G, B), H defines an equivalence relation on G with n classes, each of which is in B since the following A-universal sentence is true in algebras from graphs: ∀x : G ∃B : B ∀y : G(y ∈ B ↔ H(x, y)).Lemma 7.5.Let M = (A, G, B) be an algebra-graph system such that M |= Θ.Let X be an equivalence class of H. Then there is an ultrafilter ν of B that contains X but contains no independent sets.Proof.Let ν 0 = {B ∈ B | X − B is independent}.Then ν 0 contains X (clearly), and has the finite intersection property: Suppose for a contradiction that for B 0 , . . ., B k−1 ∈ ν 0 we have So X is the union of k independent sets in B. Now in any structure M (Γ), if an Hclass is the union of k independent sets in B, then copies of these sets for every H-class lie in B, so that Γ is the union of nk independent sets in B -that is, M (Γ) |= ¬θ nk .This implication is A-universal, so it holds in M .Hence, M |= θ nk , a contradiction.Thus ν 0 has the finite intersection property and, by the boolean prime ideal theorem, it can be extended to an ultrafilter ν, which contains X but no independent set (because it contains the complement).
Remark.The converse of Lemma 7.5 also holds, but is not needed here.Proposition 7.6.Let M = (A, G, B) be an algebra-graph system.If M |= Θ, then A is representable as a polyadic equality algebra.
Proof.By Lemma 7.4 it is sufficient to show that player ∃ has a winning strategy in the game G(A).Suppose we are in round t and the current polyadic ultrafilter network is N t .According to the rules, player ∀ chooses a ∈ A, i < n and v ∈ n N t with c i a ∈ N t (v).The other player ∃ now has to respond with a network N t+1 ⊇ N t that contains some tuple w ∈ n N t+1 such that v ≡ i w and a ∈ N t+1 (w).If there is already such a w in n N t then she can just respond with the unchanged network N t .So we assume in the following that there is no such w.
Step 1.Let N t+1 = N t ∪ {z}, where z ∈ N t is a new node.Let the tuple w be defined by w ≡ i v and w i = z.We will first try to find an ultrafilter of A for w.To help ∃ win the game, the ultrafilter should contain a.We achieve this by showing that the following set has the finite intersection property: and therefore by definition of ultrafilter networks, In algebras from graphs (and in cylindric algebras generally) we certainly have Hence, by the generalisation technique, a ∈ N t (v ).But this contradicts our assumption that no suitable tuple w exists in n N t .So we must have c i (a • D) ∈ N t (v) as claimed.
Now, if µ 0 failed the finite intersection property, there would be b , a contradiction.Thus µ 0 has the finite intersection property.
By the boolean prime ideal theorem, player ∃ can choose an ultrafilter µ of A that contains µ 0 .By construction, N t (v) ≡ i µ.Moreover, Using the definitions and Lemma 5.1, as required.
So by Lemma 6.9, N t+1 is a polyadic ultrafilter network.We also have N t+1 ⊇ N t , w ≡ i v, and a ∈ µ = N t+1 (w).The network N t+1 is ∃'s response to ∀'s move in round t.So she is able to respond to any move made by ∀ -she has a winning strategy.
1. Fix a universal axiomatisation Π of RPEA n -such an axiomatisation exists because RPEA n is a variety (Proposition 2.8).Also fix any first-order axiomatisation ∆ of RDf n .We regard Π and ∆ as A-sorted L AGS -theories in the obvious way.
2. Let Φ be the following L AGS -theory, expressing that B is infinite: 3. Also recall from Definition 7.1 that Θ = {θ k | k < ω} expresses that G has infinite chromatic number in the B-sense.The theory U defining algebra-graph systems was laid down in Definition 4.4.
We now obtain the main result of this section.It generalises the analogous result for algebras from graphs in [11].Proof.Immediate from Propositions 7.2 and 7.6.

Applications
Here we apply Theorem 7.8 to prove our two main theorems.Definition 8.1.For L Dfn ⊆ L ⊆ L PEAn , we write RL for the class of L-algebras having a representation respecting all the L-operations.

Strongly representable atom structures
Definition 8.2.Let L Dfn ⊆ L ⊆ L PEAn .An L-atom structure S is said to be strongly representable if S + ∈ RL.
The following generalises the main result of [11] to other signatures.It has already been proved by Sahed Ahmed (draft of untitled monograph, 2010) using the same algebras.
Theorem 8.3.For any L Dfn ⊆ L ⊆ L PEAn , the class of strongly representable Latom structures is non-elementary.In another common notation, the class Str RL of structures for RL is non-elementary.
Proof.A celebrated result of Erdős [4] shows that for all k < ω there is a finite graph G k with chromatic number and girth (length of the shortest cycle) both at least k.Let Γ k be the disjoint union of the G for k ≤ < ω: this time, no edges are added between copies.Plainly, Γ k has infinite chromatic number, and its girth is at least k.By Proposition 7.6 applied to M (Γ k ), A(Γ k ) L ∈ RL, so that (At Γ k ) L + is strongly representable.Now let Γ be a non-principal ultraproduct of the Γ k .Then Γ is infinite, and by Łoś's theorem it has girth at least k for all finite k, since this property is first-order definable.Hence, Γ has no cycles, so its chromatic number is at most two.By Proposition 7.2, the diagonal-free reduct of A(Γ) is not representable, and hence neither is its L-reduct.So (At Γ) L + is not strongly representable.
But it is easily seen that the operation At(−) commutes with ultraproducts, and it follows that (At Γ) L + is isomorphic to an ultraproduct of the (At Γ k ) L + .This shows that the class of strongly representable L-atom structures is not closed under ultraproducts and so cannot be elementary.

Canonical axiomatisations
Here, we use direct and inverse systems to build a certain algebra, and apply the results from the previous sections to show that it can be made to satisfy an arbitrary number of representability axioms, while its canonical extension only satisfies a bounded number.It will follow that any first-order axiomatisation of the representable cylindric algebras (and various other classes) has infinitely many non-canonical axioms.
Our argument is based on the following result.It is from [12,Lemma 4.1], but it can be proved in a rather simpler way by modifying the argument of [8,Theorem 4].Both proofs use similar random graphs.First, a definition.Definition 8.4.Let Γ, ∆ be graphs.A map f : Γ → ∆ is said to be a graph p-morphism if for each x ∈ Γ, f maps the set of neighbours of x in Γ surjectively onto the set of neighbours of f (x) in ∆.where the f ij are surjective graph p-morphisms, such that χ(Γ s ) = k for every s < ω, and χ(lim ← − Γ s ) = .
Our algebras are constructed from atom structures based on graphs, so we need to transform graph p-morphisms into p-morphisms of atom structures, and then, using duality, to embeddings of algebras.We will also consider direct and inverse systems, and their limits.Definition 8.6.Let L ⊇ L BA be a functional signature and let S = (S, R f | f ∈ L \ L BA ) and S = (S , R f | f ∈ L \ L BA ) be L-atom structures.Let g : S → S .be a function.We say that g : S → S is a p-morphism of atom structures if for each n-ary f ∈ L \ L BA , we have: Forth: g is an L + -homomorphism: for every x 1 , . . ., x n , y ∈ S, if R f (x 1 , . . ., x n , y) then R f (g(x 1 ), . . ., g(x n ), g(y)).
Our first lemma is straightforward.
For (i), suppose (K, ∼) ∈ At(Γ) and |n/∼| = n.Clearly the domain of K is preserved by f .Moreover, since im K is not independent and f × is a graph p-morphism, im K is not independent either.The other cases follow directly from the definition of f .
To show (ii) let (K , ∼) ∈ At(∆).If K is not defined anywhere, we let K be undefined everywhere as well.If there are i < j < n such that i ∼ j and K (i) = K (j) is defined, then as f × is surjective, there is p ∈ Γ × n such that f × (p) = K (i).Define K(i) = K(j) = p and let K be undefined for the remaining values in that case.Finally, if K is defined on all values i < n, then im(K ) is not independent, so there are i < j < n such that there is an edge from K (i) to K (j).Since f × is surjective, there is p i ∈ Γ × n such that f × (p i ) = K (i).As f × is a graph p-morphism, there is p j ∈ Γ × n such that there is an edge between p j and p i and f × (p j ) = K (j).For the remaining s = i, j, using surjectivity we take any vertices p s ∈ Γ × n such (i) At(G) is an inverse system of atom structures and surjective p-morphisms, (ii) A(G) is a direct system of BAOs and embeddings, (iii) A(G) + is an inverse system of atom structures and surjective p-morphisms, and For part (v), by part (iv) and (iii) we have It is clear that At(−) commutes with inverse limits, so that lim ← − At(G) ∼ = At(lim ← − G).
We can now prove the main result of the paper.

Definition 4 . 1 .
We denote by L AGS the signature with three sorts (A, G, B) and the following symbols:1.A-sorted copies of the function symbols 0, 1, +, −, d ij , c i , s σ of L PEAn for each i, j < n and σ : n → n (with the obvious arities that make A into a polyadic equality-type algebra); 2. B-sorted copies of the function symbols 0, 1, +, − of L BA ; 3. a binary (graph edge) relation symbol E on G; 4. a binary relation symbol H on G; 5. a binary relation symbol ∈ between the elements of G and B; 6. a unary function symbol R i : A → B for each i < n; 7. a unary function symbol S i : B → A for each i < n.
and since both sides are ultrafilters of A, they are equal.(iv)( =⇒ ) Assume µ ≡ i ν.For each j, k = i, we have d jk ∈ µ ⇒ d jk = c i d jk ∈ ν, and −d jk ∈ µ ⇒ −d jk = c i −d jk ∈ ν (theseequations are easily established by the generalisation technique or using basic properties of cylindric algebras: see, e.g.,[9, 1.3.3,1.2.12] and as b • D ∈ ν, we have c i a ∈ ν as well.So µ ≡ i ν by definition.(v) Immediate from (iv).

Theorem 8 . 5 .
Suppose that 2 ≤ ≤ k < ω.Then there exists an inverse system of finite graphs