A sufficient condition for finite decidability

We prove that a varietyV which is locally finite, finitely generated, congruence permutable and of finite type, and whose subdirectly irreducible algebras are all either abelian or linear type 3 above the monolith is finitely decidable if and only if the theory of the finite abelian algebras inV is decidable.Presented by R. McKenzie.     

Natural dualities for varieties of BL-algebras

BL-algebras are the Lindenbaum algebras for Hájek's Basic Logic, just as Boolean algebras correspond to the classical propositional calculus. The finite totally ordered BL-algebras are ordinal sums of MV-chains. We develop a natural duality, in the sense of Davey and Werner, for each subvariety generated by a finite BL-chain, and we use it to describe the injective and the weak injective members of these classes. The preliminary research for this paper was carried out while the second author was visiting Salerno University. The second author would like to thank the first author and Salerno University for their hospitality. The second author acknowledges partial supports from Salerno University and from the belgian Fonds National de la Recherche Scientifique.     

Free MVn-algebras

In this note we characterize free algebras in varieties of MV-algebras generated by a finite chain L _n as algebras of continuous functions from the spectrum of the Boolean skeleton of the free algebra into L _n . Received May 9, 2006; accepted in final form May 15, 2007.     

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).     

Decomposability of free Tarski algebras

In this paper we study the direct decomposability of free Tarski algebras. We show that infinite freely generated Tarski algebras are directly indecomposable, whereas finite freely generated Tarski algebras can only be decomposed into a direct product of two factors, one of which is the two-element Tarski algebra.     

IDEALS OF τ-MEASURABLE OPERATORS

Abstract

It is proven that, in general, the free algebras of an equational class, considered as an abstract category, are not definable strictly in the language of categories. As a concrete counterexample, a categorical equivalence between the categories of 2-rings and 3-rings is constructed without the axiom of choice. Isomorphism follows, as well as the non-correspondence of free algebras. An assortment of similar negative results and two open questions close the paper.     

First order logic without equality on relativized semantics

Let $\alpha \ge 2$ be any ordinal. We consider the class ${\mathsf{Drs}}_{\alpha }$ of relativized diagonal free set algebras of dimension α. With same technique, we prove several important results concerning this class. Among these results, we prove that almost all free algebras of ${\mathsf{Drs}}_{\alpha }$ are atomless, and none of these free algebras contains zero-dimensional elements other than zero and top element. The class ${\mathsf{Drs}}_{\alpha }$ corresponds to first order logic, without equality symbol, with α-many variables and on relativized semantics. Hence, in this variation of first order logic, there is no finitely axiomatizable, complete and consistent theory.     

The Variety of Near-Rings is Generated by Its Finite Members

We show that the variety of near-rings and the variety of zero-symmetric near-rings are both generated by their finite members. We show this in a more general context: if a variety is generated by a class of algebras , then the variety of -composition algebras is generated by the class of all full function algebras on direct products of finitely many copies of algebras in .     

Locally finite varieties of Heyting algebras

We show that for a variety of Heyting algebras the following conditions are equivalent: (1) is locally finite; (2) the -coproduct of any two finite -algebras is finite; (3) either coincides with the variety of Boolean algebras or finite -copowers of the three element chain are finite. We also show that a variety of Heyting algebras is generated by its finite members if, and only if, is generated by a locally finite -algebra. Finally, to the two existing criteria for varieties of Heyting algebras to be finitely generated we add the following one: is finitely generated if, and only if, is residually finite. Received November 11, 2001; accepted in final form July 25, 2005.     

Conservative extension of polyadic MV-algebras to polyadic pavelka algebras

In this paper we prove polyadic counterparts of the Hájek, Paris and Shepherdson's conservative extension theorems of Łukasiewicz predicate logic to rational Pavelka predicate logic. We also discuss the algebraic correspondents of the provability and truth degree for polyadic MV-algebras and prove a representation theorem similar to the one for polyadic Pavelka algebras.     

Varieties of two-dimensional cylindric algebras II

In [2] we investigated the lattice (Df₂) of all subvarieties of the variety Df₂ of two-dimensional diagonal free cylindric algebras. In the present paper we investigate the lattice (CA₂) of all subvarieties of the variety CA₂ of two-dimensional cylindric algebras. We prove that the cardinality of (CA₂) is that of the continuum, give a criterion for a subvariety of CA₂ to be locally finite, and describe the only pre locally nite subvariety of CA2. We also characterize nitely generated subvarieties of CA2 by describing all fteen pre nitely generated subvarieties of CA₂. Finally, we give a rough picture of (CA₂), and investigate algebraic properties preserved and reected by the reduct functors .     

Hyers-Ulam-Rassias stability of homomorphisms in quasi-Banach algebras

In this paper, we prove the Hyers-Ulam-Rassias stability of homomorphisms in quasi-Banach algebras. This is applied to investigate isomorphisms between quasi-Banach algebras.     

Nonrepresentable relation algebras generated by functional elements

There are nonrepresentable relation algebras generated by their functional elements. This solves a problem posed many years ago. The number of generating functional elements can be as low as 2. This leaves open the problem whether there is a nonrepresentable relation algebra generated by a single functional element.In Celebration of the Sixtieth Birthday of Ralph N. McKenzieReceived November 2, 2001; accepted in final form September 30, 2003.     

A characterization of free generating sets in MV-algebras

MV-algebras are a generalization of Boolean algebras. As is well known, a free generating set for a Boolean algebra is characterized by the following simple algebraic condition: whenever A and B are finite disjoint subsets of X then . Our aim in this note is to give a similar characterization of free generating sets in MV-algebras. Received January 30, 2005; accepted in final form March 13, 2007.     

Semigroups of isometries,Toeplitz algebras and twisted crossed products

We study theC ^*-algebras generated by projective isometric representations of semigroups, using a dilation theorem and the stucture theory of twisted crossed products. These algebras include the Toeplitz algebras of noncommutative tori recently studied by Ji, and similar algebras associated to the twisted group algebras of other groups such as the integer Heisenberg group.     

Decomposability of free Łukasiewicz implication algebras

?ukasiewicz implication algebras are {→,1}-subreducts of Wajsberg algebras (MV-algebras). They are the algebraic counterpart of Super-?ukasiewicz Implicational logics investigated in Komori, Nogoya Math J 72:127–133, 1978. The aim of this paper is to study the direct decomposability of free ?ukasiewicz implication algebras. We show that freely generated algebras are directly indecomposable. We also study the direct decomposability in free algebras of all its proper subvarieties and show that infinitely freely generated algebras are indecomposable, while finitely free generated algebras can be only decomposed into a direct product of two factors, one of which is the two-element implication algebra.     

On Vaught’s Conjecture and finitely valued MV algebras

We show that the complete first order theory of an MV algebra has $2^{\aleph _0}$ countable models unless the MV algebra is finitely valued. So, Vaught's Conjecture holds for all MV algebras except, possibly, for finitely valued ones. Additionally, we show that the complete theories of finitely valued MV algebras are $2^{\aleph _0}$ and that all ω‐categorical complete theories of MV algebras are finitely axiomatizable and decidable. As a final result we prove that the free algebra on countably many generators of any locally finite variety of MV algebras is ω‐categorical.