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 ℋ is Green’s
relation and ES is lexicographically ordered, S∕ℋ≅(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.
