scholarly journals A finite basis theorem for residually finite, congruence meet-semidistributive varieties

2000 ◽  
Vol 65 (1) ◽  
pp. 187-200 ◽  
Author(s):  
Ross Willard
Keyword(s):  
Finite Algebra ◽  
Finite Basis ◽  
Finitely Based ◽  
Mal’Cev Condition ◽  
Residually Finite ◽  
Basis Theorem ◽  
Finite Language

AbstractWe derive a Mal'cev condition for congruence meet-semidistributivity and then use it to prove two theorems. Theorem A: if a variety in a finite language is congruence meet-semidistributive and residually less than some finite cardinal, then it is finitely based. Theorem B: there is an algorithm which, given m < ω and a finite algebra in a finite language, determines whether the variety generated by the algebra is congruence meet-semidistributive and residually less than m.

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.

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.

Varieties generated by 2-testable monoids

2012 ◽  
Vol 49 (3) ◽  
pp. 366-389 ◽  
Author(s):  
Edmond Lee
Keyword(s):  
Present Article ◽  
Chain Condition ◽  
Finite Basis ◽  
Finitely Based ◽  
Finitely Generated ◽  
Finite Basis Problem ◽  
Basis Problem ◽  
Descending Chain Condition ◽  
Ascending Chain Condition

The smallest monoid containing a 2-testable semigroup is defined to be a 2-testable monoid. The well-known Brandt monoid B21 of order six is an example of a 2-testable monoid. The finite basis problem for 2-testable monoids was recently addressed and solved. The present article continues with the investigation by describing all monoid varieties generated by 2-testable monoids. It is shown that there are 28 such varieties, all of which are finitely generated and precisely 19 of which are finitely based. As a comparison, the sub-variety lattice of the monoid variety generated by the monoid B21 is examined. This lattice has infinite width, satisfies neither the ascending chain condition nor the descending chain condition, and contains non-finitely generated varieties.

scholarly journals A proof of Lyndon's finite basis theorem

1980 ◽  
Vol 29 (3) ◽  
pp. 229-233 ◽  
Author(s):  
Joel Berman
Keyword(s):  
Finite Basis ◽  
Basis Theorem

The finite basis property of a certain semigroup of upper triangular matrices over a field

2016 ◽  
Vol 15 (09) ◽  
pp. 1650177 ◽  
Author(s):  
Yuzhu Chen ◽  
Xun Hu ◽  
Yanfeng Luo
Keyword(s):  
Basis Property ◽  
Finite Basis ◽  
Main Diagonal ◽  
Finitely Based ◽  
Triangular Matrices ◽  
Upper Triangular Matrices ◽  
Finite Basis Property ◽  
Open Question ◽  
Nonfinitely Based ◽  
Secondary Diagonal

Let [Formula: see text] be the semigroup of all upper triangular [Formula: see text] matrices over a field [Formula: see text] whose main diagonal entries are [Formula: see text]s and/or [Formula: see text]s. Volkov proved that [Formula: see text] is nonfinitely based as both a plain semigroup and an involution semigroup under the reflection with respect to the secondary diagonal. In this paper, we shall prove that [Formula: see text] is finitely based for any field [Formula: see text]. When [Formula: see text], this result partially answers an open question posed by Volkov.

Integer points on curves of genus 1

1970 ◽  
Vol 67 (3) ◽  
pp. 595-602 ◽  
Author(s):  
A. Baker ◽  
J. Coates
Keyword(s):  
Finite Number ◽  
Finite Basis ◽  
Rational Approximations ◽  
Algebraic Numbers ◽  
Integer Points ◽  
Basis Theorem

1. Introduction. A well-known theorem of Siegel(5) states that there exist only a finite number of integer points on any curve of genus ≥ 1. Siegel's proof, published in 1929, depended, inter alia, on his earlier work concerning rational approximations to algebraic numbers and on Weil's recently established generalization of Mordell's finite basis theorem. Both of these possess a certain non-effective character and thus it is clear that Siegel's argument cannot provide an algorithm for determining all the integer points on the curve. The purpose of the present paper is to establish such an algorithm in the case of curves of genus 1.

An extension of Willard?s Finite Basis Theorem: Congruence meet-semidistributive varieties of finite critical depth

2005 ◽  
Vol 52 (2-3) ◽  
pp. 289-302 ◽  
Author(s):  
Kirby A. Baker ◽  
George F. McNulty ◽  
Ju Wang
Keyword(s):  
Finite Basis ◽  
Critical Depth ◽  
Basis Theorem

SOME NON-FINITELY BASED VARIETIES OF GROUPS AND GROUP REPRESENTATIONS

1995 ◽  
Vol 05 (03) ◽  
pp. 343-365 ◽  
Author(s):  
C.K. GUPTA ◽  
A.N. KRASIL’NIKOV
Keyword(s):  
Polynomial Identity ◽  
Finite Basis ◽  
Multilinear Polynomial ◽  
Finitely Based ◽  
Group Representations ◽  
Characteristic 2

The following results are established: (i) There exists a subvariety of the variety of centre-by-abelian -by- (nilpotent of class 2) groups which is not finitely based; (ii) There exists a variety of group representations (over a field of characteristic 2) which satisfies a multilinear polynomial identity but without any finite basis for its identities: (iii) Over fields of characteristic 2, a product of two Specht varieties of group representations need not be Specht.

THE RESIDUAL BOUND OF A FINITE ALGEBRA IS NOT COMPUTABLE

1996 ◽  
Vol 06 (01) ◽  
pp. 29-48 ◽  
Author(s):  
RALPH MCKENZIE
Keyword(s):  
Finite Type ◽  
Turing Machine ◽  
Finite Algebra ◽  
Residually Finite

We exhibit a construction which produces for every Turing machine [Formula: see text], an algebra [Formula: see text] (finite and of finite type) such that the Turing machine halts iff the variety generated by [Formula: see text] is residually finite.

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