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TitleCongruences on orthodox semigroups with associate subgroups
Author(s)Blyth, T. S.
Giraldes, E.
Smith, M. Paula Marques
KeywordsOrthodox semigroup
Associate subgroup
Inverse transversal
Issue date1996
PublisherCambridge University Press
JournalGlasgow Mathematical Journal
Abstract(s)If S is a regular semigroup then an inverse transversal of S is an inverse subsemigroup T with the property that |T intersection V(x)| = 1 for every x in S where V(x) denotes the set of inverses of x in S. In a previous publication [1] we considered the similar concept of a subsemigroup T of S such that |T intersection A(x)| = 1 for every x in S where A(x) = {y in S: xyx = x} denotes the set of associates (or pre-inverses) of x in S, and showed that such a subsemigroup T is necessarily a maximal subgroup Hα for some idempotent α in S. Throughout what follows, we shall assume that S is orthodox and α is a middle unit (in the sense that xαy = xy for all x, y in S). Under these assumptions, we obtained in [1] a structure theorem which generalises that given in [3] for uniquely unit orthodox semigroups. Adopting the notation of [1], we let T intersection A(x) = {x*} and write the subgroup T as Hα = {x*: x xin S}, which we call an associate subgroup of S. For every x x in S we therefore have x*α = x* = αx* and x*x** = α = x**x*. As shown in [1, Theorems 4, 5] we also have (xy)* = y*x* for all x, y in S, and e* = α for every idempotent e.
Publisher version10.1017/S0017089500031323
AccessOpen access
Appears in Collections:CMAT - Artigos em revistas com arbitragem / Papers in peer review journals

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