Vol. 18, No. 3, 1966

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A class of bisimple inverse semigroups

Ronson Joseph Warne

Vol. 18 (1966), No. 3, 563–577

The purpose of this paper is to study a certain generalization of the bicyclic semigroup and to determine the structure of some classes of bisimple (inverse) semigroups mod groups.

Let S be a bisimple semigroup and let ES denote the collection of idempotents of S. ES is said to be integrally ordered if under its natural order it is order isomorphic to I0, the nonnegative integers, under the reverse of their usual order. ES is lexicographically ordered if it is order isomorphic to I0 × I0 under the order (n,m) < (k,s) if k < n or k = n and s < m. If is Green’s relation and ES is lexicographically ordered, S∕(I0)4 under a simple multiplication. A generalization of this result is given to the case where ES is n-lexicographically ordered. The structure of S such that ES is integrally ordered and the structure of a class of S such that ES is lexicographically ordered are determimed mod groups. These constructions are special cases of a construction previously given by the author. This paper initiates a series of papers which take a first step beyond the Rees theorem in the structure theory of bisimple semigroups.

Mathematical Subject Classification
Primary: 20.92
Secondary: 06.00
Received: 2 July 1965
Published: 1 September 1966
Ronson Joseph Warne