Geodesic
Unit orthodox semigroups
Glasgow mathematical journal, Tome 24 (1983) no. 1, pp. 39-42

Voir la notice de l'article provenant de la source Cambridge University Press

Let S be a regular semigroup. Given x ∈ S, we shall say that a ∈ S is an associate of x if xax = x. The set of associates of x ∈ S will be denoted by A(x). Now suppose that S has an identity element 1. Let H1 denote the group of units of S. Then we say that u ∈ S is a unit associate of x whenever u ∈ A(x)∩Hl. In what follows we shall write U(x) = A(x)∩=H1, and we shall say that S is unit regular [1, 3] if (∀ x∈S)U(x)≠ ∅. Examples of unit regular semigroups include the full transformation semigroup on a finite set [1] and the semigroup of endomorphisms of a finite–dimensional vector space [3]. In this paper we shall be concerned with semigroups that are unit orthodox (i.e. unit regular and orthodox), and we shall describe completely the structure of those semigroups that are uniquely unit orthodox (i.e. orthodox and uniquely unit regular in the sense that, for every x∈S, the set U(x) is a singleton). It is worthy of mention that neither of the examples cited above is of this type. 
Blyth, T. S.; McFadden, R. Unit orthodox semigroups. Glasgow mathematical journal, Tome 24 (1983) no. 1, pp. 39-42. doi: 10.1017/S0017089500005036
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[1] 1.D'Alarcao, H., Factorizable as a finiteness condition, Semigroup Forum 20 (1980), 281–282. Google Scholar | DOI

[2] 2.Dubreil, P., Contribution à la théorie des demi-groupes, Mem. Acad. Sci. Inst. France (2) 63 (1941), 1–52. Google Scholar

[3] 3.Goodearl, K. R., von Neumann regular rings (Pitman, 1979). Google Scholar

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