Hyper‐Archimedean BL‐algebras are MV‐algebras

Generalizations of Boolean elements of a BL‐algebra L are studied. By utilizing the MV‐center MV(L) of L, it is reproved that an element x ∈ L is Boolean iff x ∨ x * = 1 . L is called semi‐Boolean if for all x ∈ L, x * is Boolean. An MV‐algebra L is semi‐Boolean iff L is a Boolean algebra. A BL‐algebra L is semi‐Boolean iff L is an SBL‐algebra. A BL‐algebra L is called hyper‐Archimedean if for all x ∈ L, xⁿ is Boolean for some finite n ≥ 1. It is proved that hyper‐Archimedean BL‐algebras are MV‐algebras. The study has application in mathematical fuzzy logics whose Lindenbaum algebras are MV‐algebras or BL‐algebras. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

On (∈, ∈ ∨ q)‐fuzzy filters of R0‐algebras

In this paper, we introduce the notions of (∈, ∈ ∨ q)‐fuzzy filters and (∈, ∈ ∨ q)‐fuzzy Boolean (implicative) filters in R₀‐algebras and investigate some of their related properties. Some characterization theorems of these generalized fuzzy filters are derived. In particular, we prove that a fuzzy set in R₀‐algebras is an (∈, ∈ ∨ q)‐fuzzy Boolean filter if and only if it is an (∈, ∈ ∨ q)‐fuzzy implicative filter. Finally, we consider the concepts of implication‐based fuzzy Boolean (implicative) filters of R₀‐algebras (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

A model of ZF with an infinite free complete Boolean algebra

By a theorem of Gaifman and Hales no model of ZF+AC (Zermelo-Fraenkel set theory plus the axiom of choice) contains an infinite free complete Boolean algebra. We construct a model of ZF in which an infinite free c.B.a. exists.     

σ‐short Boolean algebras

We introduce properties of Boolean algebras which are closely related to the existence of winning strategies in the Banach‐Mazur Boolean game. A σ‐short Boolean algebra is a Boolean algebra that has a dense subset in which every strictly descending sequence of length ω does not have a nonzero lower bound. We give a characterization of σ‐short Boolean algebras and study properties of σ‐short Boolean algebras. (© 2003 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

并素元有限生成格的弱直积分解

本文建立了并素元有限生成格的弱直积分解,并给出一个解决并素元生成的完全Heyting代数的直积分解问题的新方法;作为弱直积分解的应用,证明了并素元有限生成的完全Heyting代数必然同构于有限个既约的完全Heyting代数的直积,证明了并素元有限生成格是Boole代数的充要条件是它同构于某有限集的幂集格.     

ZF and boolean algebra

A generically embedded Boolean algebra is studied. Several results about infinite complete Boolean algebras are shown to be independent of ZF. This paper is dedicated to my parents in their 50th year of marriage. We wish to express our gratitude to Morris Orzech and the Algebra Group of Queens University, Kingston, Ontario, for their hospitality during the preparation of this paper. An erratum to this article is available at .     

Dense subtrees in complete Boolean algebras

We characterize complete Boolean algebras with dense subtrees. The main results show that a complete Boolean algebra contains a dense tree if its generic filter collapses the algebra's density to its distributivity number and the reverse holds for homogeneous algebras. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Implicitly equivalent universal algebras

The concept of implicit operation on pseudovarieties of semigroups goes back to Eilenberg and Schutzenberger [1]. The author in [2?C5] generalized this concept to other classes of algebras and established a connection between these operations and positively conditional termal functions in the case of uniform local finiteness of the algebras of the class in question. In this article we put forth the concept of an implicit operation for an arbitrary universal algebra, not necessarily locally finite, and establish a connection between these operations and infinite analogs of positively conditional terms, as well as ??-quasi-identities arising in the algebraic geometry of universal algebras. We also consider conditions for implicit equivalence of algebras to lattices, semilattices, and Boolean algebras.     

Elementary Equivalence of Linear Groups Over Rings with a Finite Number of Central Idempotents and Over Boolean Rings

In the present paper, we start with a criterion of elementary equivalence of linear groups over rings with just a finite number of central idempotents. Then we study elementary equivalence of linear groups over Boolean algebras. We prove that two linear groups over Boolean algebras are elementarily equivalent if and only if their dimensions coincide and these Boolean algebras are elementarily equivalent.     

Initial Chain Algebras on Pseudotrees

We investigate a new class of Boolean algebra, called initial chain algebras on pseudotrees. We discuss the relationship between this class and other classes of Boolean algebras. Every interval algebra, and hence every countable Boolean algebra, is an initial chain algebra. Every initial chain algebra on a tree is a superatomic Boolean algebra, and every initial chain algebra on a pseudotree is a minimally-generated Boolean algebra.We show that a free product of two infinite Boolean algebras is an initial chain algebra if and only if both factors are countable.     

On the Boolean algebras of definable sets in weakly o‐minimal theories

We consider the sets definable in the countable models of a weakly o‐minimal theory T of totally ordered structures. We investigate under which conditions their Boolean algebras are isomorphic (hence T is p‐ω‐categorical), in other words when each of these definable sets admits, if infinite, an infinite coinfinite definable subset. We show that this is true if and only if T has no infinite definable discrete (convex) subset. We examine the same problem among arbitrary theories of mere linear orders. Finally we prove that, within expansions of Boolean lattices, every weakly o‐minimal theory is p‐ω‐categorical. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Boolean products of R0‐algebras

In this paper, the (weak) Boolean representation of R₀‐algebras are investigated. In particular, we show that directly indecomposable R₀‐algebras are equivalent to local R₀‐algebras and any nontrivial R₀‐algebra is representable as a weak Boolean product of local R₀‐algebras (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

James sequences and Dependent Choices

We prove James's sequential characterization of (compact) reflexivity in set‐theory ZF + DC , where DC is the axiom of Dependent Choices. In turn, James's criterion implies that every infinite set is Dedekind‐infinite, whence it is not provable in ZF . Our proof in ZF + DC of James' criterion leads us to various notions of reflexivity which are equivalent in ZFC but are not equivalent in ZF . We also show that the weak compactness of the closed unit ball of a (simply) reflexive space does not imply the Boolean Prime Ideal theorem : this solves a question raised in [6]. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

The depth of ultraproducts of Boolean algebras

We show that if μ is a compact cardinal then the depth of ultraproducts of less than μ many Boolean algebras is at most μ plus the ultraproduct of the depths of those Boolean algebras. Received May 18, 2004; accepted in final form December 9, 2004.     

Fuzzy Weak Regular,Strong and Preassociative Filters in Residuated Lattices

In this paper, the notions of fuzzy weak regular, strong and preassociative filters are introduced with some properties of them investigated. In particular, under the context of Glivenko algebras, fuzzy weak regular filters and regular ones are equivalent and characterizations of Glivenko algebras are concluded by characterizations of fuzzy strong filters. At last the notion of fuzzy preassociative filters are defined, which are proved to coincide with fuzzy Boolean filters, and then some new alternative definitions of Boolean algebras are obtained by this type of filters.     

Some Boolean algebras with finitely many distinguished ideals II

We describe the countably saturated models and prime models (up to isomorphism) of the theory Th_prin of Boolean algebras with a principal ideal, the theory Th_max of Boolean algebras with a maximal ideal, the theory Th_ac of atomic Boolean algebras with an ideal such that the supremum of the ideal exists, and the theory Th_sa 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 T_X of Boolean algebras with distinguished ideals to be elementarily equivalent to a countably saturated model of T_X.     

Free algebras over Bezout domains are Sylvester domains

Let λ be a finitary geometric theory and δ its classifying topos. We prove that δ is Boolean if and only if (1) every first-order formula in the language of λ is ?-provably equivalent to a geometric formula and (2) for any finite list of varibles, x, there are, up to ?-provable equivalence, only finitely many formulas, in the language of λ with free variables among x. We use this characterization to show that, when δ is Boolean, it is an atomic topos and can be viewed as a finite coproduct of topoi of continuous G-sets for topological groups G satisfying a certain finiteness condition.     

Rudin‐Keisler Posets of Complete Boolean Algebras

The Rudin‐Keisler ordering of ultrafilters is extended to complete Boolean algebras and characterised in terms of elementary embeddings of Boolean ultrapowers. The result is applied to show that the Rudin‐Keisler poset of some atomless complete Boolean algebras is nontrivial.