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

Theories with equational forking

2002 ◽  
Vol 67 (1) ◽  
pp. 326-340 ◽  
Author(s):  
Markus Junker ◽  
Ingo Kraus
Keyword(s):  
Symmetric Relation ◽  
Equational Theories

AbstractWe show that equational independence in the sense of Srour equals local non-forking. We then examine so-called almost equational theories where equational independence is a symmetric relation.

scholarly journals POSITIVE FRAGMENTS OF RELEVANCE LOGIC AND ALGEBRAS OF BINARY RELATIONS

2010 ◽  
Vol 4 (1) ◽  
pp. 81-105 ◽  
Author(s):  
ROBIN HIRSCH ◽  
SZABOLCS MIKULÁS
Keyword(s):  
Equational Theory ◽  
Relevance Logic ◽  
Binary Relations ◽  
Finitely Based ◽  
Similarity Type ◽  
Relation Composition

We prove that algebras of binary relations whose similarity type includes intersection, union, and one of the residuals of relation composition form a nonfinitely axiomatizable quasivariety and that the equational theory is not finitely based. We apply this result to the problem of the completeness of the positive fragment of relevance logic with respect to binary relations.

A four-element algebra whose identities are not finitely based

1980 ◽  
Vol 11 (1) ◽  
pp. 255-260 ◽  
Author(s):  
Robert E. Park
Keyword(s):  
Finitely Based

A SIMPLE EXAMPLE OF A NON-FINITELY BASED SYSTEM OF POLYNOMIAL IDENTITIES

2002 ◽  
Vol 30 (10) ◽  
pp. 4851-4866 ◽  
Author(s):  
C. K. Gupta ◽  
Alexei N. Krasilnikov
Keyword(s):  
Finitely Based ◽  
Polynomial Identities
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