Pseudo-trees and Boolean algebras

We consider Boolean algebras constructed from pseudo-trees in various ways and make comments about related classes of Boolean algebras.     

Free Boolean algebras over unions of two well orderings

Given a partially ordered set P there exists the most general Boolean algebra which contains P as a generating set, called the free Boolean algebra over P. We study free Boolean algebras over posets of the form P=P₀∪P₁, where P₀, P₁ are well orderings. We call them nearly ordinal algebras.Answering a question of Maurice Pouzet, we show that for every uncountable cardinal κ there are κ² pairwise non-isomorphic nearly ordinal algebras of cardinality κ.Topologically, free Boolean algebras over posets correspond to compact 0-dimensional distributive lattices. In this context, we classify all closed sublattices of the product (ω₁+1)×(ω₁+1), showing that there are only ℵ₁ many types. In contrast with the last result, we show that there are ℵ₁² topological types of closed subsets of the Tikhonov plank (ω₁+1)×(ω+1).     

Generalizations of Boolean products for lattice-ordered algebras

It is shown that the Boolean center of complemented elements in a bounded integral residuated lattice characterizes direct decompositions. Generalizing both Boolean products and poset sums of residuated lattices, the concepts of poset product, Priestley product and Esakia product of algebras are defined and used to prove decomposition theorems for various ordered algebras. In particular, we show that FL_w-algebras decompose as a poset product over any finite set of join irreducible strongly central elements, and that bounded n-potent GBL-algebras are represented as Esakia products of simple n-potent MV-algebras.     

Boolean inequations

In this paper we consider Boolean inequations i.e. the inequations of the form f(X)≠0, where f is a Boolean function. The basic idea in this paper is: the inequation f(X)≠0 means that there exists p such that f(X)=p and p≠0. We give the formula which determines all the solutions of Boolean inequation.     

On Depth and Depth+ of Boolean Algebras

We show that the Depth⁺ of an ultraproduct of Boolean Algebras cannot jump over the Depth⁺ of every component by more than one cardinal. Consequently we have similar results for the Depth invariant. Research supported by the United States-Israel Binational Science Foundation. Publication 878 of the second author.     

Computably categorical Boolean algebras enriched by ideals and atoms

We describe computably categorical Boolean algebras whose language is enriched by one-place predicates that distinguish a finite set of ideals and atoms with respect to some ideals in this set.     

Minimally generated Boolean algebras

We introduce the class of minimally generated Boolean algebras, i.e. those algebras representable as the union of a continuous well-ordered chain of subalgebras A ₁ where A _i+1 is a minimal extension of A _i. Minimally generated algebras are closely related to interval algebras and superatomic algebras.     

Algebraic lattices and Boolean algebras

In this paper we establish several equivalent conditions for an algebraic lattice to be a finite Boolean algebra. This paper is dedicated to Walter Taylor. Received February 11, 2005; accepted in final form October 9, 2005.     

A Boolean algebra and a Banach space obtained by push-out iteration

Under the assumption that c is a regular cardinal, we prove the existence and uniqueness of a Boolean algebra B of size c defined by sharing the main structural properties that P(ω)/fin has under CH and in the ℵ₂-Cohen model. We prove a similar result in the category of Banach spaces.     

POWER STRUCTURES AND LOGIC

ABSTRACT

This paper generalizes the concept of a power alge bra to that of a power structure, and gives three application of power structures to logic.     

Weak effect algebras

Weak effect algebras are based on a commutative, associative and cancellative partial addition; they are moreover endowed with a partial order which is compatible with the addition, but in general not determined by it. Every BL-algebra, i.e. the Lindenbaum algebra of a theory of Basic Logic, gives rise to a weak effect algebra; to this end, the monoidal operation is restricted to a partial cancellative operation. We examine in this paper BL-effect algebras, a subclass of the weak effect algebras which properly contains all weak effect algebras arising from BL-algebras. We describe the structure of BL-effect algebras in detail. We thus generalise the well-known structure theory of BL-algebras. Namely, we show that BL-effect algebras are subdirect products of linearly ordered ones and that linearly ordered BL-effect algebras are ordinal sums of generalised effect algebras. The latter are representable by means of linearly ordered groups. This research was partially supported by the German Science Foundation (DFG) as part of the Collaborative Research Center “Computational Intelligence” (SFB 531).     

Comparison of Boolean algebras

In this work, some results related to superatomic Boolean interval algebras are presented, and proved in a topological way. Let x be an uncountable cardinal. To each I x, we can associate a superatomic interval Boolean algebra B _I of cardinality x in such a way that the following properties are equivalent: (i) I I x, (ii) B _I is a quotient algebra of B _J, and (iii) there is an homomorphism f from B _J into B _I such that for every atom b of B _I, there is an atom a of B _J satisfying f(a)=b. As a corollary, there are 2 ^x isomorphism types of superatomic interval Boolean algebras of cardinality x. This case is quite different from the countable one.     

Nonseparable UHF algebras I: Dixmier's problem

There are three natural ways to define UHF (uniformly hyperfinite) C^∗-algebras, and all three definitions are equivalent for separable algebras. In 1967 Dixmier asked whether the three definitions remain equivalent for not necessarily separable algebras. We give a complete answer to this question. More precisely, we show that in small cardinality two definitions remain equivalent, and give counterexamples in other cases. Our results do not use any additional set-theoretic axioms beyond the usual axioms, namely ZFC.     

The overlap algebra of regular opens

Overlap algebras are complete lattices enriched with an extra primitive relation, called “overlap”. The new notion of overlap relation satisfies a set of axioms intended to capture, in a positive way, the properties which hold for two elements with non-zero infimum. For each set, its powerset is an example of overlap algebra where two subsets overlap each other when their intersection is inhabited. Moreover, atomic overlap algebras are naturally isomorphic to the powerset of the set of their atoms. Overlap algebras can be seen as particular open (or overt) locales and, from a classical point of view, they essentially coincide with complete Boolean algebras. Contrary to the latter, overlap algebras offer a negation-free framework suitable, among other things, for the development of point-free topology. A lot of topology can be done “inside” the language of overlap algebra. In particular, we prove that the collection of all regular open subsets of a topological space is an example of overlap algebra which, under natural hypotheses, is atomless. Since they are a constructive counterpart to complete Boolean algebras and, at the same time, they have a more powerful axiomatization than Heyting algebras, overlap algebras are expected to turn out useful both in constructive mathematics and for applications in computer science.     

A superatomic Boolean algebra with few automorphisms

Assuming GCH, we prove that for every successor cardinal μ > ω₁, there is a superatomic Boolean algebra B such that |B| = 2^μ and |Aut B| = μ. Under ◊_ω1, the same holds for μ = ω₁. This answers Monk's Question 80 in [Mo]. Received: 1 January 1998 / Revised version: 18 May 1999 / Published online: 21 December 2000     

Injectives in residuated algebras

Injectives in several classes of structures associated with logic are characterized. Among the classes considered are residuated lattices, MTL-algebras, IMTL-algebras, BL-algebras, NM-algebras and bounded hoops.     

On amalgamation of reducts of polyadic algebras

Following research initiated by Tarski, Craig and Németi, and further pursued by Sain and others, we show that for certain subsets G of w, G polyadic algebras have the strong amalgamation property. G polyadic algebras are obtained by restricting the (similarity type and) axiomatization of -dimensional polyadic algebras to finite quantifiers and substitutions in G. Using algebraic logic, we infer that some theorems of Beth, Craig and Robinson hold for certain proper extensions of first order logic (without equality).     

Topological duality for Tarski algebras

In this paper we will generalize the representation theory developed for finite Tarski algebras given in [7]. We will introduce the notion of Tarski space as a generalization of the notion of dense Tarski set, and we will prove that the category of Tarski algebras with semi-homomorphisms is dually equivalent to the category of Tarski spaces with certain closed relations, called T-relations. By these results we will obtain that the algebraic category of Tarski algebras is dually equivalent to the category of Tarski spaces with certain partial functions. We will apply these results to give a topological characterization of the subalgebras. Received August 21, 2005; accepted in final form December 5, 2006.