A hierarchy of tree-automatic structures

# A hierarchy of tree-automatic structures

Olivier Finkel Equipe de Logique Mathématique
Institut de Mathématiques de Jussieu

CNRS and Université Paris 7, France.
Stevo Todorčević
###### Abstract

We consider -automatic structures which are relational structures whose domain and relations are accepted by automata reading ordinal words of length for some integer . We show that all these structures are -tree-automatic structures presentable by Muller or Rabin tree automata. We prove that the isomorphism relation for -automatic (resp. -automatic for ) boolean algebras (respectively, partial orders, rings, commutative rings, non commutative rings, non commutative groups) is not determined by the axiomatic system ZFC. We infer from the proof of the above result that the isomorphism problem for -automatic boolean algebras, , (respectively, rings, commutative rings, non commutative rings, non commutative groups) is neither a -set nor a -set. We obtain that there exist infinitely many -automatic, hence also -tree-automatic, atomless boolean algebras , , which are pairwise isomorphic under the continuum hypothesis CH and pairwise non isomorphic under an alternate axiom AT, strengthening a result of [Fin-Tod].

Automata reading ordinal words; -automatic structures; -tree-automatic structures; boolean algebras; partial orders; rings; groups; isomorphism relation; models of set theory; independence results.
\subjclass\revauthor

Finkel, Olivier \revauthorTodorčević, Stevo \twoaddressEquipe de Logique Mathématique
Institut de Mathématiques de Jussieu

CNRS and Université Paris 7, France. Department of Mathematics
University of Toronto, Toronto, Canada M5S 2E4.2

## 1 Introduction

An automatic structure is a relational structure whose domain and relations are recognizable by finite automata reading finite words. Automatic structures have very nice decidability and definability properties and have been much studied in the last few years, see [BlumensathGraedel00, BlumensathGraedel04, KNRS, NiesBSL, RubinPhd, RubinBSL]. They form a subclass of the class of (countable) recursive structures where “recursive” is replaced by “recognizable by finite automata”. Blumensath considered in [Blumensath99] more powerful kinds of automata. If we replace automata by tree automata (respectively, Büchi automata reading infinite words, Muller or Rabin tree automata reading infinite labelled trees) then we get the notion of tree-automatic (respectively, -automatic, -tree-automatic) structures. Notice that an -automatic or -tree-automatic structure may have uncountable cardinality. All these kinds of automatic structures have the two following fundamental properties. The class of automatic (respectively, tree-automatic, -automatic, -tree-automatic) structures is closed under first-order interpretations. The first-order theory of an automatic (respectively, tree-automatic, -automatic, -tree-automatic) structure is decidable.

On the other hand, automata reading words of ordinal length had been firstly considered by Büchi in his investigation of the decidability of the monadic second order theory of a countable ordinal, see [bs, Hemmer] and also [Woj, Woj2, Bedon96, BedonCarton98, Bedon2] for further references on the subject. We investigate in this paper -automatic structures which are relational structures whose domain and relations are accepted by automata reading ordinal words of length for some integer . All these structures are -tree-automatic structures presentable by Muller or Rabin tree automata.

A fundamental question about classes of automatic structures is the following: “what is the complexity of the isomorphism problem for some class of automatic structures?” The isomorphism problem for the class of automatic structures, or even for the class of automatic graphs, is -complete, [KNRS]. On the other hand, the isomorphism problem is decidable for automatic ordinals or for automatic boolean algebras, see [KNRS, RubinBSL]. Some more results about other classes of automatic structures may be found in [lics-KuskeLL10]: in particular, the isomorphism problem for automatic linear orders is not arithmetical. Hjorth, Khoussainov, Montalbán, and Nies proved that the isomorphism problem for -automatic structures is not a -set, [HjorthKMN08]. More Recently, Kuske, Liu, and Lohrey proved in [KuskeLL-CSL10] that the isomorphism problem for -automatic structures (respectively, partial orders, trees of finite height) is not even analytical, i.e. is not in any class where is an integer. In [Fin-Tod] we recently proved that the isomorphism relation for -tree-automatic structures (respectively, -tree-automatic boolean algebras, partial orders, rings, commutative rings, non commutative rings, non commutative groups) is not determined by the axiomatic system ZFC. This showed the importance of different axiomatic systems of Set Theory in the area of -tree-automatic structures.

We prove here that the isomorphism relation for -automatic (resp. -automatic for ) boolean algebras (respectively, partial orders, rings, commutative rings, non commutative rings, non commutative groups) is not determined by the axiomatic system ZFC. We infer from the proof of the above result that the isomorphism problem for -automatic boolean algebras, , (respectively, rings, commutative rings, non commutative rings, non commutative groups) is neither a -set nor a -set. We obtain that there exist infinitely many -automatic, hence also -tree-automatic, atomless boolean algebras , , which are pairwise isomorphic under the continuum hypothesis CH and pairwise non isomorphic under an alternate axiom AT (for “almost trivial”). This way we improve our result of [Fin-Tod], where we used the open coloring axiom OCA instead, in two ways:

• by constructing infinitely many structures with independent isomorphism problem, instead of only two such structures, and

• by finding such structures which are much simpler than the -tree-automatic ones, because they are even -automatic.

The paper is organized as follows. In Section 2 we recall definitions and first properties of automata reading ordinal words and of tree automata. In Section 3 we define -automatic structures and -tree-automatic structures and we prove simple properties of -automatic structures. We introduce in Section 4 some particular -automatic boolean algebras . We recall in Section 5 some results of Set Theory and recall in particular the Axiom AT (‘almost trivial’) and some related notions. We investigate in Section 6 the isomorphism relation for -automatic structures and for -tree-automatic structures. Some concluding remarks are given in Section 7.

## 2 Automata

### 2.1 \omn-Automata

When is a finite alphabet, a non-empty finite word over is any sequence , where for , and is an integer . The length of is . The empty word has no letter and is denoted by ; its length is . For , we write . is the set of finite words (including the empty word) over .

We assume the reader to be familiar with the elementary theory of countable ordinals. Let be a finite alphabet, and be an ordinal; a word of length (or -word) over the alphabet is an -sequence (or sequence of length ) of letters in . The set of -words over the alphabet is denoted by . The concatenation of an -word and of a -word is the -word such that for and for ; it is denoted or simply .

We assume that the reader is familiar with the notion of Büchi automaton reading infinite words over a finite alphabet which can be found for instance in [Thomas90, Staiger97]. Informally speaking an -word over is accepted by a Büchi automaton iff there is an infinite run of on enterring infinitely often in some final state of . The -language accepted by the Büchi automaton is the set of -words accepted by . A Muller automaton is a finite automaton equipped with a set of accepting sets of states. An -word over is accepted by a Muller automaton iff there is an infinite run of on such that the set of states appearing infinitely often during this run is an accepting set of states, i.e. belongs to . It is well known that an -language is accepted by a Büchi automaton iff it is accepted by a Muller automaton.

We shall define -automatic structures as relational structures presentable by automata reading words of length , for some integer . In order to read some words of transfinite length greater than , an automaton must have a transition relation for successor steps defined as usual but also a transition relation for limit steps. After the reading of a word whose length is a limit ordinal, the state of the automaton will depend on the set of states which cofinally appeared during the run of the automaton. These automata have been firstly considered by Büchi, see [bs, Hemmer]. We recall now their definition and behaviour.

###### Definition 2.1 ([Woj, Woj2, Bedon96])

An ordinal Büchi automaton is a sextuple , where is a finite alphabet, is a finite set of states, is the initial state, is the transition relation for successor steps, and is the transition relation for limit steps.

A run of the ordinal Büchi automaton reading a word of length , is an ()-sequence of states defined by: and, for , and, for a limit ordinal, , where is the set of states which cofinally appear during the reading of the first letters of , i.e.

 Inf(x,i)={q∈Q∣∀μ

A run of the automaton over the word of length is called successful if . A word of length is accepted by if there exists a successful run of over . We denote the set of words of length which are accepted by . An -language is a regular -language if there exists an ordinal Büchi automaton such that .

An ordinal Büchi automaton is said to be deterministic iff is in fact the graph of a function from into and is the graph of a function from into . In that case there is at most one run of the automaton over a given word .

###### Remark 2.2

When we consider only finite words, the language accepted by an ordinal Büchi automaton is a rational language. If we consider only -words, the -languages acceped by ordinal Büchi automata are the -languages accepted by Muller automata and then also by Büchi automata.

###### Definition 2.3

An -automaton is an ordinal Büchi automaton reading only words of length for some integer .

We can obtain regular -languages from regular -languages and regular -languages by the use of the notion of substitution. The following result appeared in [Hemmer] and has been also proved in [Finkel-loc01].

###### Proposition 2.4

Let be an integer. An -language is regular iff it is obtained from a regular -language by substituting in every -word a regular -language to each letter .

We now recall some fundamental properties of regular -languages.

###### Theorem 2.5 (Büchi-Siefkes, see [bs, Hemmer, Bedon96])

Let be an integer. One can effectively decide whether the -language accepted by a given -automaton is empty or not.

###### Theorem 2.6

[see [Bedon96]] Let be an integer. The class of regular -languages is effectively closed under finite union, finite intersection, and complementation, i.e. we can effectively construct, from two -automata and , some -automata , , and , such that , , and is the complement of .

We assume the reader to be familiar with basic notions of topology that may be found in [Moschovakis80, Kechris94, PerrinPin]. The usual Cantor topology on is the product topology obtained from the discrete topology on the finite set , for which open subsets of are in the form , where .

Let be an integer. Let be a recursive bijection. Then we have a bijection from onto defined by for each integer . Then for each -language we have the associated -language . Consider now a regular -language . It is stated in [DFR] that is Borel (in the class ).

### 2.2 Tree automata

We introduce now languages of infinite binary trees whose nodes are labelled in a finite alphabet .

A node of an infinite binary tree is represented by a finite word over the alphabet where means “right” and means “left”. Then an infinite binary tree whose nodes are labelled in is identified with a function . The set of infinite binary trees labelled in will be denoted . A tree language is a subset of , for some alphabet . (Notice that we shall only consider in the sequel infinite trees so we shall often use the term tree instead of infinite tree).

Let be a tree. A branch of is a subset of the set of nodes of which is linearly ordered by the prefix relation and which is closed under this prefix relation, i.e. if and are nodes of such that and then .
A branch of a tree is said to be maximal iff there is not any other branch of which strictly contains .

Let be an infinite binary tree in . If is a maximal branch of , then this branch is infinite. Let be the enumeration of the nodes in which is strictly increasing for the prefix order. The infinite sequence of labels of the nodes of such a maximal branch , i.e. is called a path. It is an -word over the alphabet .

For a tree and , we shall denote the subtree defined by for all . It is in fact the subtree of which is rooted in .

We are now going to define tree automata and regular languages of infinite trees.

###### Definition 2.7

A (nondeterministic) tree automaton is a quadruple , where is a finite set of states, is a finite input alphabet, is the initial state and is the transition relation. A run of the tree automaton on an infinite binary tree is an infinite binary tree such that:
(a) and   (b) for each , .

A Muller (nondeterministic) tree automaton is a 5-tuple , where is a tree automaton and is the collection of designated state sets. A run of the Muller tree automaton on an infinite binary tree is said to be accepting if for each path of , the set of states appearing infinitely often on this path is in . The tree language accepted by the Muller tree automaton is the set of infinite binary trees such that there is (at least) one accepting run of on . A tree language is regular iff there exists a Muller automaton such that .

We now recall some fundamental closure properties of regular tree languages.

###### Theorem 2.8 (Rabin, see [Rabin69, Thomas90, 2001automata, PerrinPin])

The class of regular tree languages is effectively closed under finite union, finite intersection, and complementation, i.e. we can effectively construct, from two Muller tree automata and , some Muller tree automata , , and , such that , , and is the complement of .

## 3 Automatic structures

Notice that one can consider a relation , where , are finite alphabets, as an -language over the product alphabet . In a similar way, we can consider a relation , as a tree language over the product alphabet .

Let now be a relational structure, where is the domain, and for each   is a relation of finite arity on the domain . The structure is said to be -automatic (respectively, -tree-automatic) if there is a presentation of the structure where the domain and the relations on the domain are accepted by -automata (respectively, by Muller tree automata), in the following sense.

###### Definition 3.1 (see [Blumensath99])

Let be a relational structure, where is an integer, and each relation is of finite arity .
An -tree-automatic presentation of the structure is formed by a tuple of Muller tree automata , and a mapping from onto , such that:

1. The automaton accepts an equivalence relation on , and

2. For each , the automaton accepts an -ary relation on such that is compatible with , and

3. The mapping is an isomorphism from the quotient structure
onto .

The -tree-automatic presentation is said to be injective if the equivalence relation is just the equality relation on . In this case and can be omitted and is simply an isomorphism from onto . A relational structure is said to be (injectively) -tree-automatic if it has an (injective) -tree-automatic presentation.

Notice that sometimes an -tree-automatic presentation is only given by a tuple of Muller tree automata , i.e. without the mapping . In that case we still get the -tree-automatic structure which is in fact equal to up to isomorphism.

We get the definition of -automatic (injective) presentation of a structure and of -automatic structure by simply replacing Muller tree automata by -automata in the above definition.

Notice that, due to the good decidability properties of Muller tree automata and of -automata, we can decide whether a given automaton accepts an equivalence relation on and whether, for each , the automaton accepts an -ary relation on such that is compatible with .

We denote -AUT the class of -automatic structures and -tree-AUT the class of -tree-automatic structures.

We state now two important properties of automatic structures.

###### Theorem 3.2 (see [Blumensath99])

The class of -tree-automatic (respectively, -automa
-tic) structures is closed under first-order interpretations. In other words if is an -tree-automatic (respectively, -automatic) structure and is a relational structure which is first-order interpretable in the structure , then the structure is also -tree-automatic (respectively, -automatic).

###### Theorem 3.3 (see [Hodgson, Blumensath99])

The first-order theory of an -tree-automatic (respectively, -automatic) structure is decidable.

Notice that -tree-automatic (respectively, -automatic) structures are always relational structures. However we can also consider structures equipped with functional operations like groups, by replacing as usually a -ary function by its graph which is a -ary relation. This will always be the case in the sequel where all structures are viewed as relational structures.

Some examples of -automatic structures can be found in [RubinPhd, NiesBSL, KNRS, KhoussainovR03, BlumensathGraedel04, KuskeLohrey, HjorthKMN08, KuskeLL-CSL10].

A first one is the boolean algebra of subsets of .

The additive group is -automatic, as is the product .

Assume that a finite alphabet is linearly ordered. Then the set of -words over the alphabet , equipped with the lexicographic ordering, is -automatic.

Is is easy to see that every (injectively) -automatic structure is also (injectively) -tree-automatic. Indeed a Muller tree automaton can easily simulate a Büchi automaton on the leftmost branch of an infinite tree.

The inclusions -AUT -AUT, , are straightforward to prove.

###### Proposition 3.4

For each integer ,     -AUT -tree-AUT.

{proof}

We are first going to associate a tree to each -word in such a way that if is a regular -language then the tree language will be also regular. We make this by induction on the integer . Let then be a finite alphabet and be a distinguished letter in . We begin with the case . If is an -word over the alphabet then is the tree in such that for every integer and for every word such that . It is clear that if is a regular -language then the tree language is a regular set of trees. Assume now that we have associated, for a given integer , a tree to each -word in such a way that if is a regular -language then the tree language is also regular. Consider now an -word over . It can be divided into subwords , , of length . By induction hypothesis to each -word is associated a tree . Recall that we denote by the subtree of which is rooted in . We can now associate to the -word the tree which is defined by: for every integer , and for every integer . Let then now be a regular -language. By Proposition 2.4 the language is obtained from a regular -language by substituting in every -word a regular -language to each letter . By induction hypothesis for each letter there is a tree automaton such that . This implies easily that one can construct, from a Büchi automaton accepting the regular -language and from the tree automata , , another tree automaton such that .

The inclusion -AUT -tree-AUT holds because any element of the domain of an -automatic structure, represented by an -word , can also be represented by a tree . The relations of the structure are then also presentable by tree automata.

Notice that the strictness of the inclusion follows easily from the existence of an -tree-automatic structrure without Borel presentation, proved in [HjorthKMN08], and the fact that every -automatic structrure has a Borel presentation (see the end of Section 2.1).

On the other hand we can easily see that the inclusion -AUT -AUT is strict by considering ordinals. Firstly, Kuske recently proved in [Kuske10] that the -automatic ordinals are the ordinals smaller than . Secondly, it is easy to see that the ordinal is -automatic. The ordinal is the order-type of finite sequences of integers ordered by (1) increasing length of sequences and (2) lexicographical order for sequences of integers of the same length . A finite sequence of integers can be represented by the following -word over the alphabet :

 αx=(an1+1⋅b\om)⋅(an2+1⋅b\om)⋯(anp+1⋅b\om)⋅(b\om)⋅(b\om)⋯

it is then easy to see that there is an -automaton accepting exactly the -words of the form for a finite sequence of integers . Moreover there is an -automaton recognizing the pairs such that .

## 4 Some \omn-automatic boolean algebras

We have seen that the boolean algebra of subsets of is -automatic. Another known example of -automatic boolean algebra is the boolean algebra of subsets of modulo finite sets. The set of finite subsets of is an ideal of , i.e. a subset of the powerset of such that:

1. and .

2. For all , it holds that .

3. For all , if and then .

For any two subsets and of we denote their symmetric difference. Then the relation defined by: “ iff the symmetric difference is finite” is an equivalence relation on . The quotient denoted is a boolean algebra. It is easy to see that this boolean algebra is -automatic, see for example [KuskeLohrey, HjorthKMN08, Fin-Tod].

More generally we now consider the boolean algebras for integers . We first give the definition of the sets . For we denote the order type of as a suborder of the order . The set is defined by:

 I\omn={P⊆\omn∣o.t.(P)<\omn}.

For each integer the set is an ideal of .

For any two subsets and of we denote their symmetric difference. Then the relation defined by: “ iff the symmetric difference is in ” is an equivalence relation on . The quotient , also denoted , is a boolean algebra.

We are going to show that this boolean algebra is -automatic.

We first notice that each set can be represented by an -word over the alphabet by setting if and only if for every ordinal . Let then

 Ln={xP∈{0,1}\omn∣P∈I\omn}.
###### Theorem 4.1

Let be an integer. Then the set is a regular -language.

{proof}

We reason by induction on the integer . Firstly it is easy to see that is the set of -words over the alphabet having only finitely many letters . It is a well known example of a regular -language, see [PerrinPin, Thomas90]. Notice that its complement is then also regular since the class of regular -languages is closed under complementation.

We now assume that we have proved that for each integer the set is a regular -language. In particular the language is a regular -language. Moreover its complement is then also regular since the class of regular -languages is closed under complementation.

Consider now a set . It is easy to see that belongs to if and only if there are only finitely many integers such that has order type .

Thus the -language is obtained from the regular -language by substituting the -language to the letter and the -language to the letter . We can conclude, using Proposition 2.4, that the -language is regular.

We can now state the following result.

###### Theorem 4.2

For every integer the boolean algebra is -automatic.

{proof}

Let be an integer. We denote , or simply when there is no confusion from the context, the equivalence class of a set for the equivalence relation . Let and and for any , . Then it follows easily from the preceding Theorem 4.1 that is accepted by an -automaton.

The operations , of intersection, union, and complementation, on are defined by: , , and , see [Jech].

Thus the operations of intersection, union, (respectively, complementation), considered as ternary relations (respectively, binary relation) are also given by regular -languages. On the other hand, is the equivalence class of the empty set and is the class of .

This proves that the structure is -automatic.

Notice that, as in the above proof, we can see that the relation is a regular -language because the “almost inclusion” relation is defined by: iff . Thus we can also state the following result.

###### Theorem 4.3

For each integer the structure is -automatic.

From now on we shall denote . The boolean algebra is -automatic hence also -tree-automatic.

Recall now the definition of an atomless boolean algebra.

###### Definition 4.4

Let be a boolean algebra and be the inclusion relation on defined by iff for all . Then the boolean algebra is said to be an atomless boolean algebra iff for every such that there exists an element such that .

We can now recall the following known result.

###### Proposition 4.5

For each integer the boolean algebra is an atomless boolean algebra.

{proof}

Let be an integer. Consider the boolean algebra . Let be such that the equivalence class is different from the element in . Then the set has order type and it can be splitted in two sets and such that and both and have still order type . The element is different from the element in because has order type , and because has order type . Thus the following strict inclusions hold in : . This proves that the boolean algebra is atomless.

The following result is also well known but we give a proof for completeness.

###### Proposition 4.6

For each integer the boolean algebra has the cardinality of the continuum.

{proof}

By recursion on we define a partition of into a sequence of nonempty finite sets such that for every infinite the union does not belong to the ideal For let For set where for each we have fixed a decomposition of the interval of ordinals into a sequence of finite nonempty pairwise disjoint sets with the property that has order type for every infinite

Clearly is a complete Boolean algebra embedding which induces also an embedding of into Since has cardinality continuum the conclusion follows.

In the sequel we shall often identify the powerset of a countable set with the Cantor space . Then can be equipped with the standard metric topology obained from this identification, and the topological notions like open, closed, , Borel, analytic, can be applied to families of subsets of .

###### Remark 4.7

In fact a similar result holds for an arbitrary -ideal of subsets of More precisely, by a well-known result of Talagrand ( [Tal] ; Théorème 21), for every proper -ideal of subsets of there is a partition of into a sequence of nonempty finite sets such that for every infinite the union does not belong to the ideal Therefore, as above, there is an embedding of into

###### Remark 4.8

Recall that two subsets and are said to be almost disjoint if their intersection is finite. Recall that while a countable index set does not admit an uncountable family of pairwise disjoint subsets it does admit an uncountable family of subsets that are pairwise almost disjoint. So fix an uncountable family of pairwise almost disjoint infinite subsets of For fix a sequence of nonempty finite subsets of such that for every infinite the union does not belong to the ideal (see the proof of Proposition 4.6). For let Then is an uncountable family of infinite subsets of which is also almost disjoint (i.e., for in ) but it has the additional property that for all

## 5 Axioms of set theory

We now recall some basic notions of set theory which will be useful in the sequel, and which are exposed in any textbook on set theory, like [Jech].

The usual axiomatic system ZFC is Zermelo-Fraenkel system ZF plus the axiom of choice AC. A model (V, of the axiomatic system ZFC is a collection V of sets, equipped with the membership relation , where “” means that the set is an element of the set , which satisfies the axioms of ZFC. We shall often say “ the model V” instead of “the model (V, ”.

The axioms of ZFC express some natural facts that we consider to hold in the universe of sets.

The infinite cardinals are usually denoted by

We recall that Cantor’s Continuum Hypothesis states that the cardinality of the continuum is equal to the first uncountable cardinal . Gödel and Cohen have proved that the continuum hypothesis CH is independent from the axiomatic system ZFC. This means that, assuming ZFC is consistent, there are some models of ZFC + CH and also some models of ZFC + CH, where CH denotes the negation of the continuum hypothesis, [Jech].

If V is a model of ZF and is the class of constructible sets of V, then the class forms a model of ZFC + CH.

Recall also that denotes the Open Coloring Axiom (or Todorcevic’s axiom as it is called in the more recent literature; see, for example, [Fa3]), a natural alternative to that has been first considered by the second author in [Todorcevic89]. It is known that if the theory ZFC is consistent, then so are the theories (ZFC + CH) and (ZFC + OCA), see [Jech, pages 176 and 577]. In particular, if V is a model of and if is the class of constructible sets of V, then the class forms a model of .

The axiom OCA was used in our previous paper [Fin-Tod] on tree-automatic structures but here we shall use another related axiom first considered by Just [Ju1]. To introduce this axiom we need some definitions.

Let and be two infinite countable sets, a function and an ideal of containing all finite subsets of but not the whole set . Then the function is said to preserve intersections modulo whenever

1. for every such that and

2. for every

Recall that one can identify the powerset , where is a countable set, with the set equipped with the Cantor topology which is the product topology of the discrete topology on . Thus one can also use notions like open, closed, Borel, analytic, for ideals of , where is an integer.

Then Just’s axiom (where the shorthand stands for ‘Almost Trivial’) states that for every -ideal of subsets of , for every which preserves intersections modulo , and for every uncountable family of pairwise almost disjoint infinite subsets of there exist and a finite decomposition such that for every there is a continuous function111Continuity here is interpreted when we make the standard identification of and with the Cantor cubes and respectively. such that for every

The axiom implies also the following form which will be used in the sequel: For every -ideal of subsets of , for every which preserves intersections modulo , and for every uncountable family of pairwise almost disjoint infinite subsets of there exist and a finite decomposition such that for every there is a continuous function such that for every

In [Ju1], Just showed that every model V of ZFC admits a forcing extension satisfying ZFC + AT. In another paper ([Ju2]) he showed that OCA implies many instances of AT. In particular, it is shown in [Ju2] that OCA implies AT restricted to the class of all -ideals of subsets of This was later extended by Farah [Fa] to a larger class of -ideals of subsets of , however it is still not known if OCA implies the full AT. The motivation behind the axiom AT came from the theory of quotient Boolean algebras of the form where is a proper (i.e., ) ideal on which we always assume to include the ideal of all finite subsets of Let denotes the natural quotient map, i.e. whenever A homomorphism

 Φ:P(\om)/I→P(\om)/J

between two such quotient Boolean algebras is usually given by its lifting i.e., a map for which the following diagram

 P(ω)\lx@stackrelH⟶   P(ω)↓πI      ↓πJP(ω)/ I\lx@stackrelΦ⟶  P(ω)/J

commutes. Note that any such lifting preserves intersections modulo the range ideal Note also that in general does not need to be a Boolean algebra homomorphism. It is therefore quite natural to ask for conditions on the given ideals and on and the homomorphism that would guarantee the existence of liftings that preserve the Boolean algebra operations of the algebra , even the infinitary ones. Such liftings are called completely additive liftings. Note that such completely additive liftings are always given by maps in such a way that

 H({n})=h−1(n) for all n<\om.

It follows that for all and so from this we can conclude that every completely additive lifting is a continuous map when we make the natural identification of with the Cantor set . Thus AT asserts the seemingly weak form of this, the local continuity of liftings between quotient algebras over -ideals of subsets of While local continuity of liftings is a matter of additional axioms of set theory, the second author (see, for example, [todorcevic98], Problem 1) has posed a problem about the natural mathematical counterpart of this asking under which conditions continuous liftings can be turned into completely additive ones. In subsequent work of Farah [Farah] and Kanovei-Reeken [KR] this conjecture has been verified for a very wide class of ideals of subsets of We shall use the following particular result from this work, which is a reformulation of [KR, Theorem 2], using the fact that a continuous homomorphism is actually completely additive. Notice that below the boolean algebras are a direct generalization of the boolean algebras and that we shall in fact only use in the sequel the case where the ordinal is an integer.

###### Theorem 5.1

(see [KR, Theorem 2]) For every countable ordinal if a homomorphism

 Φ:P(\om)→P(\omξ)/I\omξ

has a continuous lifting then it also has a completely additive lifting, or in other words, there is a map such that

 Φ(X)=[h−1(X)]I\omξ for all X⊆\om.

## 6 The isomorphism relation

We had proved in [Fin-Tod] that there exist two -tree automatic boolean algebras and such that: (1) (ZFC + CH) and are isomorphic. (2) (ZFC + OCA) and are not isomorphic. We are going to prove a similar result for the class of -automatic structures, for any integer using AT in place of OCA.

We first recall the following folklore result (see, for example, [Farah]).

###### Theorem 6.1

(ZFC + CH)   The boolean algebras , , are pairwise isomorphic.

Notice that this result is an immediate consequence of the simple fact that each of the Boolean algebras , , is -saturated. Therefore assuming CH, as the boolean algebras , , are all of cardinality a well-known Cantor’s back and forth argument will give us the isomorphisms. (The reader may find these notions in a textbook on Model Theory, like [Poizat]).

Note that if the equality for transfers easily to the existence of a map with the property that for every subset if and only if It follows that the corresponding map is a lifting of an isomorphic embedding