Congruences on orthodox semigroups with associate subgroups

Blyth, T. S. ; Giraldes, Emília ; Marques-Smith, M. Paula O.

Glasgow mathematical journal, Tome 38 (1996) no. 1, pp. 113-124

Résumé

If Sis a regular semigroup then an inverse transversal of S is an inverse subsemigroup T with the property that |T ∩ V(x)| = 1 for every x ∈ S where V(x) denotes the set of inverses of x ∈ S. In a previous publication [1] we considered the similar concept of a subsemigroup T of S such that |T ∩ A(x)| = 1 for every x ∩ S where A(x) = {y∈ S;xyx = x} denotes the set of associates (or pre-inverses) of x ∈ S, and showed that such a subsemigroup T is necessarily a maximal subgroup Ha for some idempotent α ∈ 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 ∈ 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 ∩ A(x) = {x*} and write the subgroup T as Hα = {x*;x ∈ S}, which we call an associate subgroup of S. For every x ∈ 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 ∈ S, and e* = α for every idempotent e.

Détail

DOI : 10.1017/S0017089500031323

Bibliographie

[1] 1.Blyth, T. S., Giraldes, Emilia and Marques-Smith, M. Paula O., Associate subgroups of orthodox semigroups, Glasgow Math. J. 36 (1994), 163–171. Google Scholar

[2] 2.Blyth, T. S. and Janowitz, M. F., Residuation theory (Pergamon Press, 1972). Google Scholar

[3] 3.Blyth, T. S. and McFadden, R., Unit orthodox semigroups, Glasgow Math. J. 24 (1983), 39–42. Google Scholar | DOI