Computable Isomorphisms of Boolean Algebras with Operators

Studia Logica (2012) 100: 481–496DOI: 10.1007/s11225-012-9411-1 © Springer 2012

B.Khoussainov

T.Kowalski

Computable Isomorphismsof Boolean Algebraswith Operators

Abstract. In this paper we investigate computable isomorphisms of Boolean algebras

with operators (BAOs). We prove that there are examples of polymodal Boolean algebras

with finitely many computable isomorphism types. We provide an example of a polymodal

BAO such that it has exactly one computable isomorphism type but whose expansions by

a constant have more than one computable isomorphism type. We also prove a general

result showing that BAOs are complete with respect to the degree spectra of structures,

computable dimensions, expansions by constants, and the degree spectra of relations.

Keywords: Computable isomorphism, Boolean algebra with operators, Degree spectrum.

1. Introduction

One of the central topics in computable algebra and model theory is con-cerned with the study of computable isomorphisms. Classically, we do notdistinguish between isomorphic structures, however, from a computabilitypoint of view, isomorphic structures can differ quite dramatically. A typicalexample is provided by the linear order of type ω. It has two computablecopies such that in one the successor function is computable, but it is notcomputable in the other. These are clearly classically isomorphic, but theyare not computably isomorphic.

Computable isomorphisms of structures have been studied intensely forat least four decades. A number of natural structures arising in universalalgebra, such as Boolean algebras, vector spaces, linear orders, trees, andAbelian groups have been considered. In the present paper we continuethis line of research and study computable isomorphisms of Boolean alge-bras with operators (BAOs). Our interest in these particular structures istwofold. On the one hand, BAOs are frequently encountered in universalalgebra, not least in their connection to modal logic (cf. [4] for more on thisconnection). On the other, there is a considerable body of results about com-putable isomorphisms of Boolean algebras. For example, Goncharov in [2]proves that the number of computable isomorphism types of any computable

Presented by R. Goldblatt; Received January 10, 2012

482 B. Khoussainov and T. Kowalski

Boolean algebra is either 1 or ω. So one may wonder how expanding the lan-guage of Boolean algebras affects computable isomorphisms. Investigationsof a similar nature have been carried out for example in the area of Abeliangroups and ordered Abelian groups. It has been shown in [6] that the num-ber of computable isomorphism types of any computable linearly orderedAbelian group is either 1 or ω, a fact true also in the class of computableAbelian groups.

In general, naturally arising computable structures tend to have either1 or ω computable isomorphism types, but there are examples of structureswith exactly n computable isomorphism types for any given n ∈ ω. Suchexamples can be found in the classes of groups, rings, integral domains,partial orders—structures of general algebraic interest [3]. Our goal is toshow that the class of computable Boolean algebras with operators is richfrom the computability-theoretic point of view. In particular, we constructexamples of Boolean algebras with operators with exactly n computableisomorphism types for any given n ∈ ω.

Now we give an overview of the paper. In the next two sections weintroduce basic concepts from computable model theory and the theory ofBAOs. Section 4 will be devoted to computable enumerations of families ofsets. These will play a central role in our results. In Section 5 we constructa BAO with exactly n computable isomorphism types, for any given n ∈ ω.This shows how the number of computable isomorphism types changes whenwe pass from Boolean algebras (where it can only be 1 or ω, see e.g., [2] or[10]) to Boolean algebras with operators. In Section 6, we demonstrate thatadding a single constant to a BAO with precisely 1 computable isomorphismtype can lift the number of isomorphism types of the expanded BAO to 2,and in fact to any finite k. The operators of the Boolean algebras constructedin Sections 5 and 6 are unary operations. Such BAOs are called polymodalalgebras because of their close relationship with modal logic (cf. [4]). In thelast section by giving up the condition of polymodality, that is by allowingoperators of arity greater than 1, we provide a much stronger result for theclass of BAOs. Indeed, we present a simple construction showing that theclass of BAOs is complete with respect to the degree spectra of structures,computable dimensions, expansions by constants, and the degree spectra ofrelations. Thus, our results for polymodal algebras are weaker than thoseproved in the last section for the class of BAOs in general, but they arestronger in that they produce examples with desired properties in a partic-ularly well-known class of BAOs. It is not clear whether the results of thelast section can be obtained for polymodal algebras by using coding argu-ments similar to ones presented in this paper. We tried to make the paper

Computable Isomorphisms of BAOs 483

accessible to a wide audience ranging from modal logicians to computabilitytheorists, so it is reasonably self-contained, except Section 7, where we as-sume familiarity with basics of Turing degrees. Since our main results donot depend on this section, uninterested readers may safely skip it.

2. Basics

We begin with presenting basic definitions from computable model theory.

Definition 2.1. An algebra A = 〈A; fn00 , fn1

1 , . . . 〉 is computable if A = ωand the function f(i, 〈x1, . . . , xni〉) = fni

i (x1, . . . , xni) is computable. Analgebra B is computably presentable if B is (classically) isomorphic to acomputable algebra A. In this case, any isomorphism from A onto B iscalled a computable presentation (or a computable copy) of B.

For instance, the structure 〈ω; s〉, with s being the successor function, is acomputable algebra, and Bω—the Boolean algebra generated by left-closed,right-open intervals of ω under its set-theoretical order—has a computablepresentation.

We are interested in those computable algebras which have the samecomputable isomorphism types. We formalize this as follows.

Definition 2.2. An isomorphism f from a computable algebra B to a com-putable algebra A is a computable isomorphism if f itself is a computablefunction. In this case we say that A and B have the same computable iso-morphism type.

For instance, any two computable copies of ω with successor are com-putably isomorphic. In general, two classically isomorphic computable al-gebras need not be computably isomorphic. The following definition makesthis simple observation precise.

Definition 2.3. The computable dimension of an algebra A is the number ofits computable isomorphism types. The algebra A is computably categoricalif its computable dimension is 1.

A typical example of a computably categorical algebra is any computablefinitely generated algebra. The reason is that any suitable mapping fromgenerators in one computable copy to generators in another can be extendedto a computable isomorphism. As another example we want to mentionthe following result obtained independently in [2] and [10] that characterizescomputably categorical Boolean algebras.


Theorem 2.4 (Dzgoev and Goncharov, Remmel). A Boolean algebra A iscomputably categorical if and only if A has finitely many atoms. Moreover,if A is not computably categorical then its computable dimension is ω.

Usually it is not difficult to find algebras of computable dimension ωor 1. Algebras of finite computable dimension greater than 1 are much moredifficult to come by. Goncharov was the first to construct such algebras forany given n (cf. [5]).

3. Boolean algebras with operators

Since Theorem 2.4 above solves completely the question of computable iso-morphism types of Boolean algebras, it seems natural to investigate thesame question with respect to algebras which have a Boolean algebra reduct.A class of such algebras that suggests itself is the class of BAOs. These areBoolean algebras with additional operations which distribute over join andpreserve zero in each argument. More formally, we have:

Definition 3.1. A Boolean algebra with operators (BAO) is an algebraA = 〈A;∧,∨,−, fi(i ∈ I), 0, 1〉 such that the reduct 〈A;∧,∨,−, 0, 1〉 is aBoolean algebra and each operation fi, (i ∈ I), say of arity k, satisfies thefollowing:

fi(a0, . . . , 0, . . . , ak−1) = 0fi(a0, . . . , b ∨ c, . . . , ak−1) = fi(a0, . . . , b, . . . , ak−1) ∨ fi(a0, . . . , c, . . . , ak−1)

in each argument j < k.

In the terminology of [7] the first condition is called normality and thesecond additivity. Operations that satisfy the properties above are calledoperators. Unary operators are often called modalities, because of the con-nection with modal logics. Accordingly, BAOs with one unary operator arecalled modal algebras, BAOs with more than one unary operator are calledpolymodal. Such are going to be the BAOs we construct.

We will adopt the following notational convention. If A is a Booleanalgebra, and f an operator on A, we will write 〈A; f〉, for the algebra〈A;∧,∨,−, f, 0, 1〉.

Our BAOs will arise from certain relational structures akin to unaryalgebras, by means of the construction we describe below. Let U = 〈U ; f0〉be an infinite structure with f0 a binary relation such that for each u ∈ U thef0-image of the singleton {u} is a finite nonempty set. It will be convenient


to view f0 as a multi-valued function on U and for that reason we will callstructures of this kind multi-valued unary algebras, although we are readyto give up this mouthful in favour of any better alternative. For a subset xof U we will write f0[x] for the f0-image of x. Clearly, U is a unary algebraiff f0[{u}] is a singleton for all u ∈ U . Now, let B be the Boolean algebra〈B;∪,∩,−, U, ∅〉, where B is the set of all finite and cofinite subsets of U .Then, define a unary operation f on B by putting:

f(x) =

{f0[x] if x is a finite subset of U,

U otherwise.

Further, define another unary operation g on B as follows:

g(x) =

⎧⎪⎨⎪⎩∅ if x = ∅,x− if x = {u} for some u ∈ U,

U otherwise.

Modal logicians will recognise g as the difference modality.

Lemma 3.2. The algebra A is a BAO.

Proof. We need to show that f and g are operators. To see that forf notice first that f0[∅] = ∅. Then, take x, y ⊆ U . If at least one ofx, y is cofinite, f(x ∪ y) = U = fx ∪ fy. If both x and y are finite,f(x∪y) = f0[x∪y] = f0[x]∪f0[y] = fx∪fy. Then, consider g(x∪y). Only thecase where x and y are atoms is not immediate. Suppose first x �= y. Thenx∪y is not an atom and x−∪y− = U . Thus, g(x∪y) = U = x−∪y− = gx∪gyas needed. If x = y, the claim holds trivially.

4. Computable enumerations

A standard method of constructing structures of finite computable dimensionis to code certain families of sets into these structures so that computabilitytheoretic properties of the families are reflected in computable isomorphismtypes. We need another definition.

Definition 4.1. A computable enumeration of a family F of computablyenumerable subsets of ω is a bijective mapping ν : ω → F such that the set{(i, x) : x ∈ ν(i)} is computably enumerable.

Two computable enumerations ν and μ of F are equivalent if there isa computable function f such that ν(i) = μ(f(i)). The idea behind this


definition is that from the ν-code of a set in F we can find its μ-code. Oneof the results instrumental in constructing structures of finite computabledimensions is the following theorem from [5].

Theorem 4.2 (Goncharov). For any n ≤ ω there exists a family F that hasexactly n pairwise non-equivalent computable enumerations.

It is known that any family with finitely many infinite sets (and possi-bly infinitely many finite ones) has either exactly one or ω non-equivalentcomputable enumerations. The construction of an F satisfying the theo-rem above requires sophisticated methods from computability theory and israther complicated. Fortunately, for the purpose at hand, all we need toknow is that such a family exists.

5. BAOs with finite computable dimensions

We start with a countable multi-valued unary algebra U and construct theassociated BAO A, as described at the end of section 3.

Definition 5.1. A (presentation of a) computable multi-valued unary al-gebra U is locally effective if there exists an algorithm that for any givenu ∈ U computes the cardinality of the set f0[u].

Note that any computable unary algebra is locally effective, as then bydefinition |f0[u]| = 1, for all u ∈ U . However it is not at all hard to giveexamples of computable multi-valued unary algebras that are not locallyeffective. Our next lemma exhibits the relationship between computablepresentations of U and the associated BAO A.

Lemma 5.2. The structure U has a locally effective computable presentationif and only if the algebra A has a computable presentation.

Proof. Suppose that U has a locally effective computable presentation.We can assume that this presentation is U itself, i.e., in particular, U = ω.Recall that Bω is the Boolean algebra generated by left-closed right-openintervals of ω under the natural ordering, and that Bω is isomorphic to thealgebra of finite and cofinite subsets of ω. Let B′

ω be a computable copyof Bω such that the set of atoms of Bω—denoted At(B′

ω) later on—is acomputable set. Let ϕ : ω → At(B′

ω) be a computable bijection. The unarymulti-valued function f0 on U naturally induces a computable multi-valuedfunction f ′

0 on At(B′ω) via ϕf0ϕ

−1. Now we can computably extend f ′0 to


a function f ′ as follows:

f ′(x) =

⎧⎪⎨⎪⎩

0 if x = 0,∨ni f0[ai] if x =

∨ni ai, and ai ∈ At(B′

ω), 1 ≤ i ≤ n,

1 otherwise.

Note that f ′ is a computable function, because U is locally effective. Further,define the function g′ by:

g′(x) =

⎧⎪⎨⎪⎩

0 if x = 0,

x− if x ∈ At(B′ω),

1 otherwise.

Clearly, this function is computable as well. Thus, 〈B′ω; f ′, g′〉 is a com-

putable copy of A, with the isomorphism established by naturally extendingϕ from atoms onto the whole universe.

For the other direction, assume that A has a computable copy A′. Theset U ′ of atoms of A′ is definable by the formula gx = x−, and thereforecomputable. On U ′ define a binary relation f ′

0 putting xf ′0y iff y ≤ f ′x

in A′. Clearly, 〈U ′; f ′0〉 is a computable structure isomorphic to U. Now,

by computability of A′ and the way f ′ behaves, given an x ∈ U ′ one cancompute the number of atoms a1, . . . , an ∈ U ′ with f ′x = y1 ∨ · · · ∨ yn. Bydefinition we have xf ′

0y iff y ∈ {y1, . . . , yn}, and therefore 〈U ′; f ′0〉 is also

locally effective.

The proof of the lemma above implicitly defines two transformations:τB and τU . The former transforms a given computable copy of a multi-valued unary algebra into a computable BAO; the latter transforms a givencomputable copy of a BAO into a computable multi-valued unary algebra.It turns out that these transformations preserve computable isomorphismtypes. We state this in the next lemma whose proof we leave to the reader.

Lemma 5.3. Multi-valued unary algebras U1 and U2 are computably isomor-phic if and only if τB(U1) is computably isomorphic to τB(U2). Similarly,BAOs A1 and A2 are computably isomorphic if and only if τU (A1) is com-putably isomorphic to τU (A2).

The two lemmas above show that in order to define BAOs with fi-nite computable dimensions is suffices to produce unary algebras of finitecomputable dimensions. The next lemma takes care of this. The readerwill notice that the unary algebras we construct here are not multi-valued,


in other words, they are indeed algebras. To construct our algebras, wewill make use of a family F of sets such that F has exactly n nonequiv-alent computable enumerations (cf. Theorem 4.2 above). Although F isan infinite family of infinite sets, for ease of exposition we present a fi-nite example first. Let X = {x1, . . . , xk} be a finite set. With X we as-sociate a unary algebra UX as follows. The universe UX is the disjointunion of AX = {a, c1, . . . , ck, d1, . . . , dk, e1, . . . , ek, g1, . . . , gk}, and BX ={b1,1, . . . , bx1,1, b1,2, . . . , bx2,2, . . . . . . , b1,k, . . . , bxk,k}. The unary operation fis defined on UX by putting

• f(a) = a

• f(d�) = f(e�) = c�, for each � ∈ {1, . . . , k}• f(g�) = e�, for each � ∈ {1, . . . , k}• f(c�) = b1,�, for each � ∈ {1, . . . , k}• f(bm,�) = bm+1,�, for each � ∈ {1, . . . , k} and each m ∈ {1, x� − 1}• f(bx�,�) = a, for each � ∈ {1, . . . , k}

Intuitively, we think of UX as an encoding of the set X; an illustrationis provided in Figure 1. We encourage readers to label the points in thisexample themselves to see that the encoding is unique. It is then clear howto generalize this construction for an infinite X. It is also not difficult tosee that given a computable enumeration of X, the construction of UX canbe carried out computably. Speaking informally again, in the right-handside diagram of Figure 1 there will be infinitely many branches, but all thebranches will be computably enumerated.

Lemma 5.4. For each n ∈ ω there exists a unary algebra U of computabledimension n.

Proof. We use the family F that has exactly n nonequivalent computableenumerations. Let ν : ω → F be a computable enumeration of F . Since F isa computable enumeration, there exists an effective procedure that for eachi ∈ ω constructs a unary algebra Ui that encodes the set ν(i), as in Figure 1.Namely, Ui is just UX with X = ν(i).

As it is easy to see from Figure 1, the construction guarantees that the setν(i) coincides with {t : (∃cj)f t+1(cj) = ai}. Informally, we view the elementai as the code of ν(i). Now, Ui can be computably constructed from ν(i).Notice that Ui does not have any nontrivial automorphisms.

Let Uν be an effective disjoint union of all Ui. The following list ofproperties shows that Uν is the desired algebra.


t+1

step

s

︷︸︸

︷

. . . . . . . . .c0 cj ck

ai

Figure 1. Left: algebra encoding the set {1, 2, 4}; right: algebra Ui in general.

1. For all computable enumerations ν and μ of F , the algebras Uν and Uμ

are isomorphic.

2. For all computable enumerations ν and μ of F , the algebras Uν and Uμ

are computably isomorphic iff ν is equivalent to μ.

3. From any computable copy U of Uν one can construct a computableenumeration νU of F such that νU and μ are equivalent iff UνU and Uμ

are computably isomorphic.

Property (1) is clear from the construction, as is the ‘if’ part of prop-erty (2). For its ‘only if’ part suppose Uν is computably isomorphic toUμ via ϕ. For an ai ∈ Uν with fai = ai, consider ϕ(ai). Since ϕ is anisomorphism, fϕ(ai) = ϕ(ai) and the set

{x ∈ Uν : fkx = ai for some k ∈ ω},

which in fact is the subalgebra Ui of Uν , is isomorphic to

{ϕ(x) ∈ Uμ : fkϕ(x) = φ(ai) for some k ∈ ω}.

Again, speaking informally, ϕ(ai) codes the same set in Uμ as ai in Uν .This proves that ν and μ are equivalent by means of ϕ composed with thecomputable bijection ai �→ i. Now, the second part of property (3) followsfrom property (2). For its first part, here is the procedure. Pick up anyelement of U ; iterate f until you reach an element a with fa = a; this is acode of some set from family F ; then keep enumerating U until the completepath from a back to the fork-shaped quadruple of elements has appeared.Now you know that the number t—the length of the path from the topof the fork to a, minus 1—belongs to the set coded by a. Repeating this


ad infinitum you will have enumerated all pairs (a, t) such that t belongs tothe set coded by a, precisely what a computable enumeration of F requires.

The lemmas above amount to a proof of the following result.

Theorem 5.5. For each n ∈ ω there exists a BAO A of computable dimen-sion n.

6. Adding constants

In this section we will address the question of how constants affect com-putable dimensions of BAOs. Notice first that adding a constant to a BAOresults in another BAO, so such expansions seem rather natural. The follow-ing theorem proved in [9] shows that a certain amount of decidability makesconstants irrelevant to the issue.

Theorem 6.1 (Millar). If a structure S is computably categorical and itsexistential theory is decidable, then any expansion of S by finitely manyconstants is also computably categorical.

However, without the decidability assumption the picture changes quitedramatically, as shown in [1].

Theorem 6.2 (Cholak, Goncharov, Khoussainov and Shore). For each nat-ural number k there is a computably categorical structure S such that theexpansion of S by a single constant has computable dimension k.

The proof of the above theorem proceeds by coding certain computablyenumerable families of k-tuples of sets. We will show that the theoremremains true if S is assumed to be a BAO. In fact, we will confine ourselves tothe case k = 2, the extension to any positive k being rather straightforwardgiven the constructions in [1]. We will need yet another definition.

Definition 6.3. Let F be a family of pairs of nonempty subsets of ω. Wecall F symmetric, if (A,B) ∈ F implies (B, A) ∈ F . The family F iscomputably enumerable, if there is a bijective mapping ν : ω → F such thatthe set {(i, x, y) : x ∈ lν(i), y ∈ rν(i)} is computably enumerable, where land r stand for the left and right projections of pairs.

If ν : ω → F is a computable enumeration of a symmetric family F , thenνd defined as νd(i) = (rν(i), lν(i)) is another computable enumeration of F .To prove the result we announced, we will need a symmetric family F for


which ν and νd are the only computable enumerations possible. That is thecontent of another theorem from [1].

Theorem 6.4. There is a symmetric family F and its computable enumera-tion ν such that ν is not equivalent to νd and every computable enumerationof F is equivalent to either ν of νd.

The existence of the family F granted, and given a computable enumer-ation ν of F , we will construct a multi-valued unary algebra Mν , whosefragment, showing how to encode two pairs from F , is depicted in Figure 2.A formal definition of Mν is a rather obvious modification of the definitionof the algebra UX given just before Lemma 5.4; we leave its reconstructionto the interested reader. As before, informally we say that the element ai

codes the pair (A, B) under enumeration ν, the element bj codes the samepair under νd. By symmetry, the pair (B, A) must also appear somewhere:we have drawn it as coded by aj and bj under ν and νd, respectively. Thewhole Mν is obtained by repeating this for all pairs (A,B) ∈ F , so it can bevisualized as a daisy-wheel-like structure around the elements c and d (seeFigure 2).

Given a structure S and an element s ∈ S, we will write 〈S, s〉 for thestructure of the type of S expanded by a single constant interpreted as theelement s. For instance, 〈Mν , c〉 and 〈Mν , d〉 arise from Mν by adding aname for the element c and d, respectively.

Lemma 6.5. The structure Mν has the following properties:

1. There is a single nontrivial automorphism α of Mν , and α(ai) = bj.

2. If μ is a computable enumeration of F , then Mμ is isomorphic to Mν .

3. The structures 〈Mν , c〉 and 〈Mν , d〉 are classically isomorphic, but notcomputably so.

4. The computable dimension of 〈Mν , c〉 is 2.

5. The computable dimension of Mν is 1.

Proof. Part (1) is clear from the picture; the automorphism in question isjust the flipping over. For (2) notice that μ is equivalent to either ν of νd.Therefore, Mμ is isomorphic (computably) to either Mν or Mνd . These arein turn isomorphic by (1). For (3), classical isomorphism is clear. Suppose〈Mν , c〉 and 〈Mν , d〉 are computably isomorphic. The isomorphism theninduces a computable function ai �→ bi, which makes enumerations ν andνd equivalent. This contradiction shows that such an isomorphism cannot


c

d

ai

bi

aj

bj

A

B

B

A

Figure 2. Fragment of Mν .

be computable. For (4) we have by construction that from a computablecopy 〈M, e〉 of 〈Mν , c〉 a computable enumeration μ of F can be effectivelyobtained, for instance as a listing of members of the computable set {x ∈M : x �= e, e ∈ f0[x], x ∈ f2

0 [x]}. If μ is equivalent to ν, then 〈M, e〉 iscomputably isomorphic to 〈Mν , c〉; otherwise μ is equivalent to νd, and then〈M, e〉 is computably isomorphic to 〈Mν , d〉. Finally, to prove (5) let M bea computable copy of Mν . Enumerate M until an element e with e ∈ f0[e]appears. Then send e to either c or d and then proceed as in the case withthe additional constant. This establishes a computable isomorphism.

Theorem 6.6. There is a computably categorical BAO A, such that for someelement a ∈ A, the expanded algebra 〈A, a〉 has computable dimension 2.

Proof. Take the multi-valued unary algebra Mν and produce the BAO Aas in section 3. Then pick the element c ∈ Mν ; clearly a = {c} is an element(in fact, an atom) of A. Repeating the argument from section 4 we see thatthe computable dimensions of A and 〈A, a〉 are respectively the same as thecomputable dimensions of Mν and 〈Mν , c〉.

By coding “families of dimension k”, as they are called in [1], we canalso obtain the following generalization of Theorem 6.6, which we will statewithout proof.

Theorem 6.7. For any k ∈ ω, there is a computably categorical BAO A,such that for some element a ∈ A, the expanded algebra 〈A, a〉 has com-putable dimension k.


7. Generalization

For this section we assume that the reader is familiar with the basics ofTuring degrees. For a structure A, a presentation of A is any structure Bisomorphic to A such that the domain of B is ω. The degree of the presen-tation B is the Turing degree of its atomic diagram. For a structure A letDgSp(A) be the set of degrees of all presentations of the structure. The setDgSp(A) is called the degree spectrum of A. An important concept relatedto the study of computable isomorphisms is the notion of the degree spec-trum of a relation U on a given structure A. The degree spectrum of U onA, which is denoted by DgSp(U), is the set of all Turing degrees of the im-ages of U in all computable presentations of A. We say that the relation Uin a computable structure is intrinsically computable (computably enumer-able) if all images of U in all computable presentations of the structure arecomputable (computably enumerable). Similarly, the relation U in a struc-ture A is relatively intrinsically computable (computably enumerable) if allimages of U in all presentations A of the structure are deg(A)-computable(computably enumerable). Now we give the following definition from [3]:

Definition 7.1. A class of structures X is complete with respect to degreespectra of nontrivial structures, computable dimensions, expansions by con-stants, and degree spectra of relations if for every countable graph G thereis an A in X with the following properties:

1. DgSp(G)=DgSp(A).

2. If G is computably presentable then the following hold:

(a) For any degree d, DgSp(A) has the same d-computable dimensionas DgSp(G).

(b) If g ∈ G then there is an a ∈ A such that (A, a) has the samecomputable dimension as (G, g).

(c) If U ⊂ G then there is a V ⊂ A such that DgSp(U)=DgSp(V ) andif U is intrinsically computably enumerable, then so is V .

The rationale for the definition is that all structures can be encoded ingraphs, preserving the relevant computability properties. Thus, the classof all graphs is trivially complete with respect to degree spectra of nontriv-ial structures, computable dimensions, expansions by constants, and degreespectra of relations. But it is not the only such class of structures. Formore examples, see [3]. Now we use the following general theorem provedin [3] in order to show that the class of BAOs is complete with respect to


degree spectra of nontrivial structures, computable dimensions, expansionsby constants, and degree spectra of relations.

Theorem 7.2. Let X be a class of structures. If for every countable graphG there is an AG in X and relatively intrinsically computable invariantrelations D, which is unary, and R, which is binary, satisfying propertiesP (0)–P (3) below then X is complete with respect to degree spectra of nontriv-ial structures, computable dimensions, expansions by constants, and degreespectra of relations.

(P0) For every presentation G of G, the structure AG is deg(G)-comput-able.

(P1) For every presentation G of G there is a deg(G)-computable bijectionfG : D(AG) → |G| such that RAG(x, y) ⇔ EG(fG(x), fG(y)) for everyx, y ∈ D(AG).

(P2) Any bijection of D(AG) that preserves R can be extended to an auto-morphism of AG.

(P3) For every presentation G of G there is a deg(G)-computable set ofexistential formulas φ0(a, b0, x), φ0(a, b1, x), . . . such that a is a tuple ofelements of AG, each bi is a tuple of elements of D(AG), each x in AG

satisfies some φi, and no two elements of AG satisfy the same φi.

Now we use this theorem to give the following general result about com-putability theoretic properties of BAOs.

Theorem 7.3. The class of BAOs is complete with respect to degree spectraof nontrivial structures, computable dimensions, expansions by constants,and degree spectra of relations.

Proof. Let B be a fixed computable presentation of the Boolean algebra offinite and cofinite subsets of ω, with bottom element 0 and top element 1. Wemay assume that B satisfies the following property. There is an algorithmthat given a finite or cofinite set D produces the image x of this set in Bso that the set {(n, x) : n ∈ D} is computable. Let G be a countable graphwith the universe ω and edge relation E. Define the following operationsf(x, y) and g(x) on the domain of the Boolean algebra B as follows:

f(x, y) =

{1 if there are n ∈ x and m ∈ y such that E(n,m),0 otherwise.


and

g(x) =

⎧⎪⎨⎪⎩

0 if x = 0,

x− if x is a singleton,

1 otherwise.

These two functions are clearly computable. Consider now the structureAG = (B, f, g). This structure is a BAO. Indeed, the proof that g is anoperator is similar to what we have already proved in the previous sections.For f , our argument is the following. Note that f(x, 0) = f(0, x) = 0 forall x. Clearly, if f(x, y) = 1 or f(x, z) = 1 then f(x, y ∪ z) = 1. On theother hand, if f(x, y) = f(x, z) = 0 then for all n ∈ x and m ∈ y we have¬E(n,m), and likewise for z. Hence for all n ∈ x and all m ∈ y ∪ z we have¬E(n,m). Therefore f(x, y ∪ z) = 0. The argument for f(y ∪ z, x) = 0 issymmetric.

In order to finish off the proof of the theorem, it suffices to show that themap G → AG satisfies all the properties required in the theorem above. Itis clear that AG is deg(G)-computable. We need to define D and R. DefineD = {x : g(x) = x−} and R = {(x, y) : x, y ∈ D & f(x, y) = 1}. Thesetwo sets are definable by quantifier free formulas without parameters, andtherefore are relatively intrinsically computable and invariant. It is not hardto check that the conditions of the theorem above are satisfied.

8. Closing remarks

As we mentioned in the introduction, the computable dimension of any com-putable Boolean algebra is either 1 or ω. In this paper we produced poly-modal BAOs with computable dimension k, for any k ∈ ω. Although ourconstruction employed two unary operators, there are ways of achieving thesame with only one. It suffices to say that the simulation technique from [8],enabling one to mimic the action of two operators with only one, preservescomputability. We have also shown that the class of all BAOs is completewith respect to degree spectra of nontrivial structures, computable dimen-sions, expansions by constants, and degree spectra of relations. We do notknow if these completeness results can be obtained for polymodal BAOs bycoding methods similar to ones presented in this paper. Other natural ques-tions connected to the theme of this paper are the following. It would beinteresting to find a natural variety V of modal algebras which is not term-equivalent to Boolean algebras and yet the computable dimension of anyalgebra from V is either 1 or ω, thus extending the result for Boolean alge-bras. In the variety of monadic algebras all subdirectly irreducible members


are such, so this variety may well be the first candidate to consider. Anotherpossible direction would be to study questions related to computable isomor-phisms of Boolean algebras with distinguished ideals, a topic not discussedin this paper.

Finally, we would like to thank the referee for comments which wereessential for the improvement of the initial version of this paper. The resultsof the last section are due to suggestions provided by the referee.

References

[1] Cholak, P., S. Goncharov, B. Khoussainov, and R. Shore, Computably cate-

gorical structures and expansions by constants. Journal of Symbolic Logic 64:13–37,

1999.

[2] Dzgoev, V., and S. Goncharov, Autostability of models. Algebra and Logic 19:45–

53, 1980.

[3] Hirschfeldt, D., B. Khoussainov, R. Shore, and A. Slinko, Degree spectra

and computable dimension in algebraic structures. Annals of Pure and Applied Logic

115:71–113, 2002.

[4] Goldblatt, R., Algebraic Polymodal Logic: A Survey. Logic Journal of the IGPL

8(4):393–450, 2000.

[5] Goncharov, S., The problem of the number of non-self-equivalent constructiviza-

tions. Algebra and Logic 19:621–639, 1988.

[6] Goncharov, S., S. Lempp, and R. Solomon, The computable dimension of ordered

Abelian groups. Advances of Mathematics 175:102–143, 2003.

[7] Jonsson, B., and A. Tarski, Boolean algebras with operators. Part I. American

Journal of Mathematics 73:891–939, 1951.

[8] Kracht, M., and F. Wolter, Normal modal logics can simulate all others. Journal

of Symbolic Logic 64:99–138, 1999.

[9] Millar, T., Recursive categoricity and persistence. Journal of Symbolic Logic 51:430–

434, 1986.

[10] Remmel, J., Recursive isomorphism types of recursive Boolean algebras. Journal of

Symbolic Logic 46:572–593, 1981.

Bakhadyr KhoussainovDepartment of Computer ScienceAuckland UniversityAuckland, New [email protected]

Tomasz KowalskiDepartment of Mathematics and StatisticsLa Trobe UniversityBundoora, VIC 3086, [email protected]

Documents

Computable Isomorphisms of Boolean Algebras with Operators