Infinite chains of non-finitely based equational theories of finite algebras

1981 ◽  
Vol 13 (1) ◽  
pp. 373-378 ◽  
Author(s):  
George F. McNulty
Keyword(s):  
Finitely Based ◽  
Finite Algebras ◽  
Equational Theories

Finite axiomatizability for equational theories of computable groupoids

1989 ◽  
Vol 54 (3) ◽  
pp. 1018-1022 ◽  
Author(s):  
Peter Perkins
Keyword(s):  
Binary Operation ◽  
Finite Algebra ◽  
Equational Theory ◽  
Natural Numbers ◽  
Finite Axiomatizability ◽  
Finite Algebras ◽  
Equational Theories ◽  
Primitive Recursive ◽  
Positive Integers ◽  
Recursive Set

A computable groupoid is an algebra ‹N, g› where N is the set of natural numbers and g is a recursive (total) binary operation on N. A set L of natural numbers is a computable list of computable groupoids iff L is recursive, ‹N, ϕe› is a computable groupoid for each e ∈ L, and e ∈ L whenever e codes a primitive recursive description of a binary operation on N.Theorem 1. Let L be any computable list of computable groupoids. The set {e ∈ L: the equational theory of ‹N, ϕe› is finitely axiomatizable} is not recursive.Theorem 2. Let S be any recursive set of positive integers. A computable groupoid GS can be constructed so that S is inifinite iff GS has a finitely axiomatizable equational theory.The problem of deciding which finite algebras have finitely axiomatizable equational theories has remained open since it was first posed by Tarski in the early 1960's. Indeed, it is still not known whether the set of such finite algebras is recursively (or corecursively) enumerable. McKenzie [7] has shown that this question of finite axiomatizability for any (finite) algebra of finite type can be reduced to that for a (finite) groupoid.

Various varieties

1973 ◽  
Vol 16 (3) ◽  
pp. 363-367 ◽  
Author(s):  
Sheila Oates MacDonald
Keyword(s):  
Finite Algebra ◽  
Finite Basis ◽  
Finitely Based ◽  
Universal Algebras ◽  
Finite Algebras ◽  
The Past ◽  
Bounded Number

The study of varieties of universal algebras2 which was initiated by Birkhoff in 1935, [2], has received considerable attention during the past decade; the question of particular interest being: “Which varieties have a finite basis for their laws?” In that paper Birkhoff showed that the laws of a finite algebra which involve a bounded number of variables are finitely based, so it is not altogether surprising that finite algebras have received their share of this attention.

THE RESIDUAL BOUNDS OF FINITE ALGEBRAS

1996 ◽  
Vol 06 (01) ◽  
pp. 1-28 ◽  
Author(s):  
RALPH MCKENZIE
Keyword(s):  
Simple Algebra ◽  
Finitely Based ◽  
Finite Algebras

We exhibit, for every finite cardinal λ≥3 and also for each of λ=ω, ω1, (2ω)+, a fourelement algebra that generates a precisely residually < λ variety. We exhibit an eight-element simple algebra with eight operations that is inherently non-finitely-based and generates a precisely residually countable variety.

Strongly finitely based equational theories

1991 ◽  
Vol 28 (4) ◽  
pp. 549-558 ◽  
Author(s):  
W. Rautenberg
Keyword(s):  
Finitely Based ◽  
Equational Theories

The existence of finitely based lower covers for finitely based equational theories

1995 ◽  
Vol 60 (4) ◽  
pp. 1242-1250
Author(s):  
Jaroslav Ježek ◽  
George F. McNulty
Keyword(s):  
Equational Theory ◽  
Single Equation ◽  
Isomorphism Type ◽  
Extensive Literature ◽  
Fundamental Operation ◽  
Finitely Based ◽  
Equational Theories ◽  
Set Inclusion ◽  
Universal Quantifiers

By an equational theory we mean a set of equations from some fixed language which is closed with respect to logical consequences. We regard equations as universal sentences whose quantifier-free parts are equations between terms. In our notation, we suppress the universal quantifiers. Once a language has been fixed, the collection of all equational theories for that language is a lattice ordered by set inclusion The meet in this lattice is simply intersection; the join of a collection of equational theories is the equational theory axiomatized by the union of the collection. In this paper we prove, for languages with only finitely many fundamental operation symbols, that any nontrivial finitely axiomatizable equational theory covers some other finitely axiomatizable equational theory. In fact, our result is a little more general.There is an extensive literature concerning lattices of equational theories. These lattices are always algebraic. Compact elements of these lattices are the finitely axiomatizable equational theories. We also call them finitely based. The largest element in the lattice is compact; it is the equational theory based on the single equation x ≈ y. The smallest element of the lattice is the trivial theory consisting of tautological equations. For all but the simplest languages, the lattice of equational theories is intricate. R. McKenzie in [6] was able to prove in essence that the underlying language can be recovered from the isomorphism type of this lattice.

FINITELY BASED WORDS

2000 ◽  
Vol 10 (04) ◽  
pp. 457-480 ◽  
Author(s):  
OLGA SAPIR
Keyword(s):  
Sufficient Conditions ◽  
Basis Property ◽  
Free Monoid ◽  
Finite Basis ◽  
Finitely Based ◽  
Finite Basis Property ◽  
Letter Alphabet ◽  
Finite Language ◽  
Rees Quotient ◽  
The Ideal

Let W be a finite language and let Wc be the closure of W under taking subwords. Let S(W) denote the Rees quotient of a free monoid over the ideal consisting of all words that are not in Wc. We call W finitely based if the monoid S(W) is finitely based. Although these semigroups have easy structure they behave "generically" with respect to the finite basis property [6]. In this paper, we describe all finitely based words in a two-letter alphabet. We also find some necessary and some sufficient conditions for a set of words to be finitely based.

A JUST NON-FINITELY BASED VARIETY OF BIGROUPS

2001 ◽  
Vol 29 (9) ◽  
pp. 4011-4046 ◽  
Author(s):  
C. K. Gupta* ◽  
A. N. Krasilnikov
Keyword(s):  
Finitely Based ◽  
Finitely Based Variety
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