Isomorphism of Atomless Boolean Algebras with Distinguished Ideals

For a complete theory of Boolean algebras T, let MT denote the class of countable models of T. For B1, B2 ∈ MT, let B1 ≤ B2 mean that B2 is elementarily embeddable in B2. Theorem 1. For every complete theory of Boolean algebras T, if T ≠ Tω, then ‹MT, ≤› is well-quasi-ordered. ∎ We define Tω. For a Boolean algebra B, let I(B) be the ideal of all elements of the form a + s such that B ↾ a is an atomic Boolean algebra and B ↾ s is an atomless Boolean algebra. The Tarski derivative of B is defined as follows: B(0) = B and B(n + 1) = B(n)/I(B(n)). Define Tω to be the theory of all Boolean algebras such that for every n ∈ ω, B(n) ≠ {0}. By Tarski [1949], Tω is complete. Recall that ‹A, < › is partial well-quasi-ordering, it is a partial quasi-ordering and for every {ai, ⃒ i ∈ ω} ⊆ A, there are i < j < ω such that ai ≤ aj. Theorem 2. contains a subset M such that the partial orderings ‹M, ≤ ↾ M› and are isomorphic. ∎ Let M′0 denote the class of all countable Boolean algebras. For B1, B2 ∈ M′0, let B1 ≤′ B2 mean that B1 is embeddable in B2. Remark. ‹M′0, ≤′› is well-quasi-ordered. ∎ This follows from Laver's theorem [1971] that the class of countable linear orderings with the embeddability relation is well-quasi-ordered and the fact that every countable Boolean algebra is isomorphic to a Boolean algebra of a linear ordering.

Abstract.Assume that all algebras are atomless. (1) Spind(A × B) = Spind(A) ∪ Spind(B). (2) Spind(Ai). Now suppose that κ and λ are infinite cardinals, with κ uncountable and regular and with κ < λ. (3) There is an atomless Boolean algebra A such that u(A) = κ and i(A) = λ. (4) If λ is also regular, then there is an atomless Boolean algebra A such that t(A) = s(A) = κ and α (A) = λ. All results are in ZFC, and answer some problems posed in Monk [01] and Monk [∞].

Can the content of a whole paper be explained by a single footnote In 1956 Alfred Tarski made an effort in this direction in his chapter ‘On the Foundations of Boolean Algebra’ (Tarski, 1956c). This chapter had originally appeared as the paper ‘Zur Grundlegung der Boole’schen Algebra, I’ (Tarski, 1935), the sequel of which had never been published. This sequel should have contained, among other things, the atomless system of Boolean algebra, as Tarski points out (Tarski, 1956c, p. 341, fn. 2). In that same footnote Tarski also gives a model for atomless Boolean algebra, consisting of the family of so-called regular open sets of a Euclidean space and the relation of set-inclusion. He then refers back to another chapter in Tarski (1956b): Foundations of the Geometry of Solids (Tarski, 1956a). This reference and the model of atomless Boolean algebra appear only in the 1956 edition (Tarski, 1956c); they are absent in Tarski (1935).

Within the frames of the Σ-definability approach propounded by Yu. L. Ershov, we study into the definability of Boolean algebras and their Frechet ranks in hereditarily finite superstructures. Examples are constructed of a superatomic Boolean algebra whose Frechet rank is not Σ-definable in the hereditarily finite superstructure over that algebra, and of an admissible set in which the atomless Boolean algebra is not autostable.

We considerωn-automatic structures which are relational structures whose domain and relations are accepted by automata reading ordinal words of lengthωnfor some integern≥ 1. We show that all these structures areω-tree-automatic structures presentable by Muller or Rabin tree automata. We prove that the isomorphism relation forω2-automatic (resp.ωn-automatic forn> 2) 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 forωn-automatic boolean algebras,n≥ 2, (respectively, rings, commutative rings, non commutative rings, non commutative groups) is neither a-set nor a-set. We obtain that there exist infinitely manyωn-automatic, hence alsoω-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 [14].

Recall that a subset X of an algebra A is irredundant iff x ∉ 〈X∖{x}〉 for all x ϵ X, where 〈X∖{x}) is the subalgebra generated by X∖{x}. By Zorn's lemma there is always a maximal irredundant set in an algebra. This gives rise to a natural cardinal function Irrmm(A) = min{∣X∣: X is a maximal irredundant subset of A}. The first half of this article is devoted to proving that there is an atomless Boolean algebra A of size 2ω for which Irrmm(A) = ω.A subset X of a BA A is ideal independent iff x ∉ (X∖{x}〉id for all x ϵ X, where 〈X∖{x}〉id is the ideal generated by X∖{x}. Again, by Zorn's lemma there is always a maximal ideal independent subset of any Boolean algebra. We then consider two associated functions. A spectrum functionSspect(A) = {∣X∣: X is a maximal ideal independent subset of A}and the least element of this set, smm(A). We show that many sets of infinite cardinals can appear as Sspect(A). The relationship of Smm to similar “continuum cardinals” is investigated. It is shown that it is relatively consistent that Smm/fin) < 2ω.We use the letter s here because of the relationship of ideal independence with the well-known cardinal invariant spread; see Monk [5]. Namely, sup{∣X∣: X is ideal independent in A} is the same as the spread of the Stone space Ult(A); the spread of a topological space X is the supremum of cardinalities of discrete subspaces.

In the database framework of Kanellakis et al. it was argued that constraint query languages should meet the closed-form requirement, that is, queries should take as input constraint databases and give as output constraint databases that use the same type of constraints. This paper shows that the closed-form requirement can be met for Datalog queries with Boolean equality constraints with double exponential time-complete data complexity, for Datalog queries with precedence and monotone inequality constraints in triple exponential-time data complexity. A closed-form evaluation is also shown for (Stratified) Datalog queries with equality and inequality constraints in atomless Boolean algebras in triple exponential-time data complexity.

This paper continues the study of a class of inverse monoids, called Tarski monoids, that can be regarded as non-commutative generalizations of the unique countable, atomless Boolean algebra. These inverse monoids are related to a class of étale topological groupoids under a non-commutative generalization of classical Stone duality and, significantly, they arise naturally in the theory of dynamical systems as developed by Matui. We are thereby able to reinterpret a theorem of Matui (à la Rubin) on a class of étale groupoids as an equivalent theorem about a class of Tarski monoids: two simple Tarski monoids are isomorphic if and only if their groups of units are isomorphic. The inverse monoids in question may also be viewed as countably infinite generalizations of finite symmetric inverse monoids. Their groups of units therefore generalize the finite symmetric groups and include amongst their number the Thompson groups $$V_{n}$$ .

In this paper, we investigate a new model theoretical tree property (TP), called the antichain tree property (ATP). We develop combinatorial techniques for ATP. First, we show that ATP is always witnessed by a formula in a single free variable, and for formulas, not having ATP is closed under disjunction. Second, we show the equivalence of ATP and [Formula: see text]-ATP, and provide a criterion for theories to have not ATP (being NATP). Using these combinatorial observations, we find algebraic examples of ATP and NATP, including pure groups, pure fields and valued fields. More precisely, we prove Mekler’s construction for groups, Chatzidakis’ style criterion for pseudo-algebraically closed (PAC) fields, and the AKE-style principle for valued fields preserving NATP. We give a construction of an antichain tree in the Skolem arithmetic and atomless Boolean algebras.

The nonmonotonic deduction relation in default reasoning is defined with fixed point style, which has the many-extension property that classical logic is not possessed of. These two kinds of deductions both have boolean definability property, that is, their extensions or deductive closures can be defined by boolean formulas. A generalized form of fixed point method is employed to define a class of deduction relations, which all have the above property. Theorems on definability and atomless boolean algebras in model theory are essential in dealing with this assertion.

The formalization of the “part – of” relationship goes back to the mereology of S. Leśniewski, subsequently taken up by Leonard & Goodman (1940), and Clarke (1981). In this paper we investigate relation algebras obtained from different notions of “part-of”, respectively, “connectedness” in various domains. We obtain minimal models for the relational part of mereology in a general setting, and when the underlying set is an atomless Boolean algebra.

The paper shows that the homeomorphism groups of, respectively, Cantor's discontinuum, the rationals and the irrationals have uncountable cofinality. It is well known that the homeomorphism group of Cantor's discontinuum is isomorphic to the automorphism group Aut ${\bb B}$ of the countable, atomless boolean algebra ${\bb B}$ . So also Aut ${\bb B}$ has uncountable cofinality, which answers a question posed earlier by the first author and H. D. Macpherson. The cofinality of a group $G$ is the cardinality of the length of a shortest chain of proper subgroups terminating at $G$ .

We prove that the form of conditional independence at play in database theory and independence logic is reducible to the first-order dividing calculus in the theory of atomless Boolean algebras. This establishes interesting connections between independence in database theory and stochastic independence. As indeed, in light of the aforementioned reduction and recent work of Ben-Yaacov (Isr. J. Math. 194(2):957---1012, 2013), the former case of independence can be seen as the discrete version of the latter.

Every \(\textrm{low}_n\) Boolean algebra, for 1 ≤ n ≤ 4, is isomorphic to a computable Boolean algebra. It is not yet known whether the same is true for n > 4. However, it is known that there exists a \(\textrm{low}_5\) subalgebra of the computable atomless Boolean algebra which, when viewed as a relation on the computable atomless Boolean algebra, does not have a computable copy. We adapt the proof of this recent result to show that there exists a \(\textrm{low}_4\) subalgebra of the computable atomless Boolean algebra which, when viewed as a relation on the computable atomless Boolean algebra, has no computable copy. This result provides a sharp contrast with the one which shows that every \(\textrm{low}_4\) Boolean algebra has a computable copy. That is, the spectrum of the subalgebra as a unary relation can contain a \(\textrm{low}_4\) degree without containing the degree 0, even though no spectrum of a Boolean algebra (viewed as a structure) can do the same.

We describe the countably saturated models and prime models (up to isomorphism) of the theory Thprin of Boolean algebras with a principal ideal, the theory Thmax of Boolean algebras with a maximal ideal, the theory Thac of atomic Boolean algebras with an ideal such that the supremum of the ideal exists, and the theory Thsa of atomless Boolean algebras with an ideal such that the supremum of the ideal exists. We prove that there are infinitely many completions of the theory of Boolean algebras with a distinguished ideal that do not have a countably saturated model. Also, we give a sufficient condition for a model of the theory TX of Boolean algebras with distinguished ideals to be elementarily equivalent to a countably saturated model of TX.