Residual finiteness in permutation varieties of semigroups
type: Item
instance of: scholarly article
title: Residual finiteness in permutation varieties of semigroups
zbMATH Open document ID: 0552.20041
DOI: 10.1007/BF02572638
published in: Semigroup Forum
publication date: 1985-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/134735
review text: A semigroup S is called residually finite if for all \(a\neq b\) in S there is a congruence \(\rho\) on S such that S/\(\rho\) is finite and \(\neg (a \rho b)\). In this paper the permutation varieties of semigroups, in which every finitely generated (f.g.) member is residually finite, are determined. Based on theorems of Yu. Mal'tsev, A. Ananin and T. Evans on varieties of associative algebras the following theorem is proved: If E is a set of permutation identities and if \(V_ E (M_ E)\) denotes the variety of semigroups (associative algebras) then the following are equivalent: 1) \(V_ E\) satisfies \(xy^ nx=y^ iy^{n-1-j}xy^ j\) for some \(i,j,n\in {\mathbb{Z}}^{+,0}\) with \(0<i+j\leq n\); 2) \(V_ E\) is locally representable, i.e. every f.g. member of \(V_ E\) can be embedded into a matrix algebra over a field of characteristic zero; 3) \(V_ E\) is locally Hopfian, i.e. no f.g. member of \(V_ E\) is isomorphic to a proper image; 4) \(V_ E\) is locally residually finite, i.e. every f.g. member of \(V_ E\) is residually finite; 5) \(V_ E\) is locally Noetherian, i.e. the congruence lattice of any f.g. member of \(V_ E\) satisfies the ascending chain condition; 6) \(M_ E\) is locally Noetherian; 7) Every f.g. member of \(V_ E\) is finitely presented. - Finally, the solvability of the word problem for any finite presentation of an f.g. semigroup S in \(V_ E\) satisfying (1) and thus any condition above, is proved.
Mathematics Subject Classification ID: 20M07
Mathematics Subject Classification ID: 20M05
Mathematics Subject Classification ID: 16Rxx
zbMATH DE Number: 3879163
zbMATH Keywords: residually finite
zbMATH Keywords: congruence
zbMATH Keywords: permutation varieties of semigroups
zbMATH Keywords: finitely generated
zbMATH Keywords: varieties of associative algebras
zbMATH Keywords: permutation identities
zbMATH Keywords: variety of semigroups
zbMATH Keywords: locally representable
zbMATH Keywords: locally Hopfian
zbMATH Keywords: locally Noetherian
zbMATH Keywords: congruence lattice
zbMATH Keywords: finitely presented
zbMATH Keywords: word problem
author: Thomas E. Nordahl
reviewed by: H. Mitsch
MaRDI profile type: Publication
cites work: Locally finitely approximable and locally representable varieties of algebras
cites work: New decision algorithms for finitely presented commutative semigroups
cites work: Q3880981
cites work: Residual finiteness of finitely generated commutative semigroups
cites work: Arithmetical and semihereditary semigroup rings
cites work: Rational sets in commutative monoids
cites work: Finitely Presented Loops, Lattices, etc. are Hopfian
cites work: Residually Finite Semigroups of Endomorphisms
cites work: Some Connections between Residual Finiteness, Finite Embeddability and the Word Problem
cites work: The lattice of semigroup varieties
cites work: Word problems
cites work: Shorter Notes: Redei's Finiteness Theorem for Commutative Semigroups
cites work: Q3856954
cites work: On Infinite Soluble Groups (III)
cites work: On a Theorem of Malcev
cites work: On nilpotency and residual finiteness in semigroups
cites work: On monoids presented by a single relation
cites work: Generalization of the Hilbert theorem on the finiteness of bases
cites work: A Matrix Representation for Associative Algebras. II
cites work: Residually small varieties of semigroups
cites work: A note on residually small varieties of semigroups
cites work: The decision problem for some classes of sentences without quantifiers
cites work: Q5834703
cites work: Some properties of the product of varieties of associative algebras
cites work: Varieties of associative algebras
cites work: Semigroups and their lattice of congruences. II
cites work: On the decidability of some problems in special classes of groups
cites work: Computational algorithms for deciding some problems for nilpotent groups
cites work: Finitely generated left commutative semigroups are residually finite
cites work: VARIETIES OF FINITELY APPROXIMABLE GROUPS
cites work: Residual finiteness and the Hopf property in rings
cites work: Q5679044
cites work: Q5343411
cites work: Homomorphisms and subdirect decompositions of semigroups
cites work: Residually small varieties
cites work: Finite basis theorem for systems of semigroup identities
description: scientific article; zbMATH DE number 3879163