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Principally ordered regular semigroups. The “boxing ring” and the “bootlace”. (Portuguese) Zbl 0827.06010
Proceedings of the 3rd meeting of the Portuguese algebraists, April 15-16, 1993, Coimbra, Portugal. Coimbra: Departamento de Matemática, Universidade de Coimbra, 131-139 (1993).
An ordered semigroup $$S$$ is called principally ordered when, for every $$x$$ in $$S$$, there exists a greatest element $$x^*$$ such that $$xx^* x\leq x$$. A simple argument shows that each element of a principally ordered regular semigroup possesses a greatest inverse. The author produces various examples of principally ordered semigroups and announces a few theorems about regular principally ordered semigroups from his doctoral dissertation (1992).
Reviewer’s remark. An explicit expression for $$x^*$$ and for the greatest inverse element of $$x$$, when $$x$$ is invertible, in the case of semigroups of binary relations, was given by the reviewer [Semigroup Forum 13, 95-102 (1976; Zbl 0355.20058)].
For the entire collection see [Zbl 0799.00020].
##### MSC:
 06F05 Ordered semigroups and monoids 20M17 Regular semigroups