on complete representations of algebras of logic

On complete representations ofalgebras of logic

MOHAMED KHALED AND TAREK SAYED-AHMED, Department ofMathematics, Faculty of Science, Cairo University, Giza, Egypt.E-mail: [email protected]

AbstractWe show that there exists an atomic polyadic equality algebra of dimension n that is elementary equivalent to acompletely representable algebra, but its diagonal free reduct (obtained by deleting diagonals and substitutions) isnot completely representable.

Throughout this note, n is a finite ordinal >2. The classes Dfn , CAn , PEAn , of diagonalfree cylindric algebras, cylindric algebras and polyadic equality algebras of dimension n aredefined in [4]. In [5], definition 29, a set of coloured graphs G is defined, which we now recall:

Definition 1. A coloured graph is an undirected graph � such that every edge of � is colouredby a unique edge colour (below), and some (n−1) - tuples have unique colours, too. The edgecolours are :greens: gi(i=1,...,n−2) and gi0(i<ω);whites: wi(i=0,...,n−2);reds: rmi (i<m<ω.)The colours for (n−1) - tuples are :yellows: yS (S⊆ω,S=ω or S finite).

We will sometimes write �(x,y) for the colour of an edge (x,y), and �(a1,...,an−1) for thecolour of an (n−1)-tuple a1,...,an−1 in the coloured graph �.Let �,� be coloured graphs, and ψ :�→� be a map. ψ is said to be coloured graph

embedding, or simply an embedding, if it is one to one and preserves all edges, and allcolours, where defined, in both directions.Let i<ω and let � be a coloured graph consisting of n nodes, x0,...,xn−2,y, such that(xj ,y) is an edge of � for each j<n−1. We call � an i−cone if for each j<n−1, the edge(xj ,y) is coloured gj if j>0, and gi0 if j=0, and no other edge of � (if any) are colouredgreen. The apex of the cone is y, its base {x0,...,xn−2}. The tint of the cone is i. These arewell-defined, as any � can be viewed as a cone in at most one way. Notice that a cone inducesa linear ordering on its base, namely, x0,...,xn−2.

Definition 2. The class G consists of all coloured graphs � with the following properties.

(1) � is a complete graph (all possible edges are present)(2) � contains no triangles of the following types:• (g,g ′,g∗) any green coloures g,g ′,g∗

• (gi,gi,wi) any i=1,...,n−2• (gj0,g

k0 ,w0) any j,k<ω

Vol. 17 No. 3, © The Author 2009. Published by Oxford University Press. All rights reserved.For Permissions, please email: [email protected]:10.1093/jigpal/jzp007

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268 On complete representations of algebras of logic

• (rmi ,rm′

j ,rm∗k ) unless m=m ′ =m∗ and |{i,j,k}|=3.

(3) If a1,...,an−2∈� are distinct, and no edge (ai,aj)(i< j<n−1) is coloured green, then thesequence (a1,...,an−2) is coloured a unique shade of yellow. No other (n−1) - tuples arecoloured yellow.

(4) If D={d0,...,dn−2,δ}⊆� and � �D ( the coloured graph induced on D) is an i-cone withapex δ, inducing the ordering d0,...,dn−2 on its base, and the tuple (d0,...,dn−2) is colouredyS , then i∈S.Clearly, G is closed under isomorphism and under induced subgraphs. G depends on n.In the next definition we show how these graphs can be used to construct a polyadic atomstructure.

Definition 3. Consider the class Kn of surjective maps a :n(={0,...,n−1})→�a, any �a ∈G.Many of these maps, though formally distinct, will differ only because the nodes in the imagegraphs will not be the same. So we define an equivalence relation ∼ on Kn

a∼b⇐⇒a(i)=a(j)⇔b(i)=b(j)

and

�a(a(i),a(j))=�a(b(i),b(j)),

if defined, and

�a(a(k0),...a(kn−2))=�b(b(k0)...b(kn−2)),

if defined, for all i,j ∈n and (n−1)-tuples k of elements of n. In other words, a and bdefine isomorphic coloured graphs. This is an equivalence relation on Kn. We define an atomstructure C ′

n with domain

C ′n={[a] :a∈Kn}.

For every i,j ∈n and [a],[b]∈C ′n we define Eij⊆C ′

n, Ti⊆ 2C ′n and Pij⊆ 2C ′

n as follows:

[a]∈Eij⇐⇒a(i)=a(j)

[a]Ti[b]⇐⇒a[n\{i}]∼b[n\{i}].

That is [a]Ti[b] if and only if for some c∈[a],b(j)=c(j) for all j �= i. Finally

[a]Pij [b]⇐⇒b◦[i,j]∼a

These definitions are sound (do not depend on the representatives). Now consider the atomstructure

C ′n=(C ′

n,Ti,Pij ,Eij)i,j<n .

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On complete representations of algebras of logic 269

Let Dn be the complex algebra over C ′n. That is Dn=℘(C ′

n). The boolean operations are theusual set theoretic intersections and taking complements; the extra non boolean operationsare defined for X ∈℘(C ′

n) as follows

ciX={[b]∈C ′n :∃[a]∈X [a]Ti[b]},

pijX={[b]∈C ′n :∃[a]∈X [a]Pij [b]},

dij=Eij .Let Cn to be the subalgebra of Dn generated by the atoms, i.e. by the set {[a] :a∈Kn}.Lemma 4. The algebra Cn is generated by n−1 dimensional elementsProof. It suffices to show that {[a]}=∏{ci{[a]} : i<n} for any a∈Kn . Assume that a :n→�

with �∈G. Clearly ≤ holds. For the other direction assume that b :n→� and [a] �=[b]. Weshow that b cannot be an element of the right hand side. Since a and b are not equivalent,we can assume that

1. (∃i,j<n)�(b(i),b(j)) �=�(a(i),a(j)) or2. (∃i1,...in−1<n)�(b(i1),...b(in−1)) �=�(a(i1),...a(in−1)).In the first case, let k be distinct from i and j . Then [b] /∈ck{[a]}. In the second case, choosek /∈{i1,...in−1} and proceed the same way.We shall need the notion of atomic networks which are basically finite approximations tocomplete representations.

Definition 5. Let D be an atomic arbitrary n-dimensional polyadic-type algebra. Let AtDdenotes the set of its atoms. An atomic network N is a set of nodes � and a total functionN :n�→At(D) such that− N (δ)≤dij iff δi=δj (for any δ∈n� and any i,j<n).− N (δid)≤ciN (δ) (for any i<n,δ∈n�,d ∈�).− pijN (δ)=N (δ◦[i,j].)A complete representation is a representation that preserves infinitary meets and joins when-ever defined. In [5] it is proved that an algebra has a complete representation if and only ifit has an atomic one. An atomic representation of A is a representation, i.e a map f :A→Bwhere B is a set algebra with unit nU such that nU =⋃{f (x) :x is an atom of A}. We nowprove the main result of this paper:

Theorem 6. Cn is elementary equivalent to a completely representable PEAn, hence is repre-sentable, but its Df reduct has no complete representation.

Proof. A “graph game" is defined between two players ∃ (female) and ∀ (male) in [5], cf.lemma 30. It is shown that for all k<ω, ∃ has a winning strategy in the graph game oflength k [5] proposition 33 while ∀ has a winning strategy for the graph game of length ω [5]proposition 32. Another game on networks Gk(D), D is an atomic polyadic algebra, has krounds, k≤ω, and is defined as follows. In the zero'th round, ∀ picks any atom a of D. ∃ has

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to respond with a finite atomic network N0 such that N0(d)=a for some n-tuple of nodesd ∈n0N . Without loss, |N0|≤n. In any further round, let the last network played be N . ∀ picksan index i<n, a “face" F=(f0,f1,...,fn−2)∈Nn−1, and an atom b≤ciN (f0,...fi−1,x,fi ...,fn−2)(the choice of x ∈N is arbitrary, as the second part of the definition of an atomic networktogether with the fact that ci(cix)=cix for all x ∈D ensures that the right-hand side doesnot depend on it). ∃ must respond, if possible, with a network N ⊆N+ with at most onemore node, such that there is a node d ∈N+ with N+(f0,...,fi−1,d,fi ...,fn−2)=b. If she can dothis in every round, she has won the play. It is proved in [5], lemma 31, that ∃ has a winningstrategy in the graph games of all finite lengths if and only if she has a winning strategy inthe games Gk(Cn) and that ∃ has a winning strategy in the graph game of length ω if andonly if she has a winning strategy in the game Gω(Cn). Cn is clearly countable. Now for allk<ω, ∃ has a winning strategy σk in Gk(Cn). Let B be a non-principal ultrapower of Cn . Then∃ has a winning strategy σ in Gω(B), essentially she uses σn in the nth component of theultraproduct so that at each round ofGω(B) ∃ is still winning in co-finitely many components,this suffices to show that she has still not lost. Now we can use an elementary chain argumentto construct countable elementary subalgebras of B containing Cn . Cn=A0≤A1 ...≤B. Forthis let Ai+1 be a countable elementary subalgebra of B containing Ai and all elements of Bthat σ selects in play of Gω(B) in which ∀ only chooses elements from Ai . Let A′ =⋃

i∈ωAi .This is a countable elementary subalgebra of B and ∃ has a winning strategy in Gω(A′). Weprove that A′ has a complete representation, this will prove that Cn is representable (sincethe class of representable algebras is a variety and Cn≡A′.) Consider a play N0⊆N1⊆ ... ofGω(A′) in which ∃ uses her strategy, so that all the Nt are atomic networks, and ∀ eventuallypicks up every face (f0,...fn−2) every i<n and every atom b. That is ∀ plays every possiblelegal move in some stage of the play. He can do this because there are countably many nodesthat appear in the play and countably many atoms in A′. If ∃ uses her winning strategyto this particular game, then the limit network M =⋃

t<ω(Nt) will satisfy the followingcondition (*)For every face (f0,...fn−2)∈n−1M for all i<n for every atom b of A if b≤ciM (f0,...,x,fn−2),then there exists a node l such that b=M (f0,...l,...fn−2). For r ∈A′, let

h(r)={d ∈nM :∃t<ω(d ∈nNt and Nt(d)≤r)}.

For every d ∈nM there is a t<ω and an atom a=Nt(d) such that d ∈h(Nt(d)), so that h isan atomic representation. We check the boolean operations. We have d ∈h(r+s) iff ∃t<ω,such that d ∈nNt and Nt(d)≤r+s. Because Nt(d) is an atom this is equivalent to ∃t<ω,d ∈nNt and Nt(d)≤r or Nt(d)≤s. Equivalently s∈h(r)∪h(s). Complement is just as easy:d ∈h(−r) iff d /∈h(r). Indeed assume that d ∈h(−r). Then ∃t1<ω, Nt1 (d)≤−r . d cannot bein h(r) for else we get t2<ω such that Nt2 (d)≤r . By taking t=max{t1,t2} it follows thatNt(d)=0 which is impossible because Nt(d) is an atom. The reverse inclusion is the same.Now h preserves cylindrifications by (*). Indeed, the following follows from (*): there existst1<ω such that Nt1 (d)≤cir iff there exists t2∈ω such that Nt2 (d iu)≤r for some u. Now wecheck substitutions. Let d ∈h(pij r). Then there exists t<ω such that Nt(d)≤pij r . HencepijNt(d)=Nt[d ◦[i,j])≤r . The other inclusion is similar. Preserving diagonals follows fromthe definition of atomic networks. Now we show that the diagonal free reduct of Cn has nocomplete representation. Assume, seeking a contradiction, that it has a complete representa-tion. Then since Cn is generated by n−1 dimensional elements, we have by theorems 5.1.51and 5.4.26 in [4] that Cn as a polyadic equality algebra has a complete representation h. Then

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On complete representations of algebras of logic 271

∃ can use h as a guide and win Gω(Cn), which contradicts the above. Let the base of h be U .Any finite subset N ⊆U defines an atomic network by defining N (d) to be the unique atoma∈Cn such that d ∈h(a). Such an atom exists since the representation is atomic. It is easyto see that, so defined, N is a network. ∃ ensures that each network is played this way. Fort=0 let ∀ choose an atom a. ∃ chooses d ∈nU with d ∈h(a). She defines the atomic networkby stating that its notes are d0,d1 ...dn . Given the inductive hypothesis Nt⊆U at round t, ∀chooses (f0,...fn−2)∈n−1Nt , i<n and atom b≤ciNt(f0,x,...fn−2). Then (f0,x,...fn−2)∈h(cib).Hence there exists z ∈U such that (f0,z,...fn−2)∈h(b). ∃ selects such a z and forms Nt+1 bystating that its nodes are those of Nt together with z . This completes the proof.

It follows that not only the class of completely representable cylindric algebra is not elemen-tary as proved in [5], but for any class with signature between Dfn and PEAn the class ofcompletely representable algebras is not elementary.Remark We note that the Df part of our theorem follows directly from the result in [5],

together with lemma 4, and theorem 5.1.51 in [4]. Indeed in [5] Cn is proved to be elementaryequivalent to A such that A has a complete representation but Cn does not. Let Rddf standfor the diagonal free reduct. So if one takes the diagonal free reducts, then Rddf Cn is stillelementary equivalent to RddfA, the latter has a complete representation but the formerdoes not by theorem 5.1.51. One thus might be tempted to think that the PEA part of ourtheorem follows from [5] together with lemma 4 and theorem 5.4.26 in [4]. But this is nottrue. Let us see why. In this case we need to expand the cylindric algebras of [5] to bepolyadic algebras. But this done in the definition of Cn above. However we have not provedthat Cn is actually a PEAn , all we know (before Theorem 6) is that it has the similarity typeof PEAn , the fact that it is a PEAn and indeed a representable one follows from the proofof Theorem 6. Even if we prove that Cn is a PEAn before Theorem 6, which is not a trivialmatter (but can be done), all we know is that it does not have a complete representation asa PEAn . But then how shall we find an A∈PEAn such that A≡Cn and A has a completerepresentation. [5] provides us only with a CAn with the required properties, and we do notknow anything about the generators of this CAn , nor does the proof therein shows that itis a reduct of a PEAn . In other words, we cannot directly use Theorem 5.4.26. So it seemsto us that the longer proof of Theorem 6 is required to prove our result. Besides Theorem6 shows that any class of algebras with signature between Dfn and PEAn is not elementary.Other algebras of logic to which our result applies are Pinter's substitution algebras (SC ),and Halmos polyadic algebras (PA).Using the above one can easily show that there is a simple countable representable atomicPEAn whose Df reduct is not completely representable. Such a result can be used to provethat both the omitting types theorem and Vaught's theorem on the existence of atomicmodels [2] fail for finite variable fragments of first order logic as long as the number ofvariables >2 [1]. In contrast we have every atomic representable A∈PEA2 is completelyrepresentable. We sketch a proof. By theorem 3.2.65 in [4] A has no defective atoms, since itis representable. Define Dat , small, big, Aab as in lemma 3.2.59. For a∈Dat, let Xa={(a,i) :i<µ} where µ=|AtA|+ω. Let U =⋃

a∈DatXa . A mapping φ from AtA to ℘(2U ) is definedby defining φ �Aab in [4] lemma 5.4.3. Then for any x ∈A, let

f (x)=⋃

{φ(a) :a≤x,a∈Dat}.Then clearly f is an atomic representation. The preservation of all operations other than thediagonal elements can be done as in theorem 5.4.32. The preservation of diagonal elements

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can be done by adapting the corresponding part in lemma 3.2.59. On the other hand, usingthe techniques in lemma 3.2.59, theorem 3.2.65, lemma 5.1.46, theorem 5.4.33 in [4], it canbe proved without much difficulty that for n≤2 and K ∈{Df ,CA,PA,PEA,SC } the classof completely representable algebras in Kn is elementary. In fact this class coincides withthe class of atomic representable algebras. In the cases of Df ,PA,SC the completely repre-sentable algebras are, like boolean algebras, exactly the atomic algebras. For K ∈{CA,PEA}atomic algebras can contain defective atoms under the diagonal element, cf. [4] lemma 3.2.59,that prevents building an ordinary representation in which the unit is a disjoint union ofcartesian squares. Representable algebras, however, do not contain such defective atoms.To summarize we have:

Theorem 7. Let n∈ω. Let K be a signature between Dfn and PEAn. Then the class of com-pletely representable K algebras is elementary if and only if n≤2, in which case this classcoincides with the (elementary) class of atomic representable algebras.

We believe that Theorem 7 is a significant addition to the results in [5]. The if part is neweven for cylindric algebras.

References[1] Andreka, H., Nemeti, I. Sayed Ahmed T., Omitting types for finite variable frag-ments and complete representations of algebras. Journal of Symbolic Logic 73(1) (2008)p.65–89.

[2] Chang.C, Keisler. J Model Theory, North Holland, 1994.[3] Henkin,L., Monk,J.D., and Tarski, A., Cylindric Algebras Part I. North Holland, 1971.[4] Henkin,L., Monk,J.D., and Tarski,A., Cylindric Algebras Part II. North Holland, 1985.[5] Hirsch,R., and Hodkinson,I., Complete representations in algebraic logic, Journal ofSymbolic Logic vol 62 , no 3, (1997), p.816–884.

Received 15 October 2008

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