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Congruences on orthodox semigroups with associate subgroups

  • T. S. Blyth (a1), Emília Giraldes (a2) and M. Paula O. Marques-Smith (a3)

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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 xS 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 |TA(x)| = 1 for every xS where A(x) = {yS;xyx = x} denotes the set of associates (or pre-inverses) of xS, 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*;xS}, which we call an associate subgroup of S. For every xS 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, yS, and e* = α for every idempotent e.

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References

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1.Blyth, T. S., Giraldes, Emilia and Marques-Smith, M. Paula O., Associate subgroups of orthodox semigroups, Glasgow Math. J. 36 (1994), 163171.
2.Blyth, T. S. and Janowitz, M. F., Residuation theory (Pergamon Press, 1972).
3.Blyth, T. S. and McFadden, R., Unit orthodox semigroups, Glasgow Math. J. 24 (1983), 3942.

Congruences on orthodox semigroups with associate subgroups

  • T. S. Blyth (a1), Emília Giraldes (a2) and M. Paula O. Marques-Smith (a3)

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