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Universal varieties of semigroups

Part of: Algebraic structures Semigroups Special categories

Published online by Cambridge University Press:  09 April 2009

V. Koubek  and
J. Sichler
V. Koubek
Affiliation:
MFF KU Malostranské nám. 25 Praha 1 Czechoslovakia
J. Sichler
Affiliation:
Department of Mathematics University of Manitoba Winnipeg, ManitobaCanadaR3T 2N2
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Abstract

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A category V is called universal (or binding) if every category of algebras is isomorphic to a full subcategory of V. The main result states that a semigroup variety V is universal if and only if it contains all commutative semigroups and fails the identity xnyn = (xy)n for every n ≥ 1. Further-more, the universality of a semigroup variety V is equivalent to the existence in V of a nontrivial semigroup whose endomorphism monoid is trivial, and also to the representability of every monoid as the monoid of all endomorphisms of some semigroup in V. Every universal semigroup variety contains a minimal one with this property while there is no smallest universal semigroup variety.

MSC classification

Secondary: 18B15: Embedding theorems, universal categories 20M07: Varieties and pseudovarieties of semigroups 20M15: Mappings of semigroups 08A35: Automorphisms, endomorphisms
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 36 , Issue 2 , April 1984 , pp. 143 - 152
Copyright
Copyright © Australian Mathematical Society 1984

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