How to construct finite algebras which are not finitely based

1985 ◽  
pp. 167-174 ◽  
Author(s):  
George F. McNulty
Keyword(s):  
Finitely Based ◽  
Finite Algebras

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.

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

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

scholarly journals On representing Mn's by congruence lattice of finite algebras

1983 ◽  
Vol 44 (3) ◽  
pp. 299-308 ◽  
Author(s):  
M.G. Stone ◽  
R.H. Weedmark
Keyword(s):  
Congruence Lattice ◽  
Finite Algebras

On Pseudovarieties of Forest Algebras

2016 ◽  
Vol 27 (08) ◽  
pp. 909-941 ◽  
Author(s):  
Saeid Alirezazadeh
Keyword(s):  
Natural Extension ◽  
Direct Products ◽  
Property A ◽  
Residually Finite ◽  
Finite Algebras

Forest algebras are defined for investigating languages of forests [ordered sequences] of unranked trees, where a node may have more than two [ordered] successors. They consist of two monoids, the horizontal and the vertical, with an action of the vertical monoid on the horizontal monoid, and a complementary axiom of faithfulness. In the study of forest algebras one of the main difficulties is how to handle the faithfulness property. A pseudovariety is a class of finite algebras of a given signature, closed under the taking of homomorphic images, subalgebras and finitary direct products. We tried to adapt in this context some of the results in the theory of semigroups, specially the studies on relatively free profinite semigroups, which are an important tool in the theory of pseudovarieties of semigroups. We define a new version of syntactic congruence of a subset of the free forest algebra, not just a forest language. This new version is the natural extension of the syntactic congruence for monoids in the case of forest algebras and is used in the proof of an analog of Hunter’s Lemma. We show that under a certain assumption the two versions of syntactic congruences coincide. We adapt some results of Almeida on metric semigroups to the context of forest algebras. We show that the analog of Hunter’s Lemma holds for metric forest algebras, which leads to the result that zero-dimensional compact metric forest algebras are residually finite. We show an analog of Reiterman’s Theorem, which is based on a study of the structure profinite forest algebras.

Sign in / Sign up

Export Citation Format

Share Document