The biorder on the partial groupoid of idempotents of a semigroup

The fundamental four-spiral semigroup

The class CS of completely simple semigroups forms a variety under the operations of multiplication and inversion (x−1 being the inverse of x in its ℋ-class). We determine a Rees matrix representation of the CS-free product of an arbitrary family of completely simple semigroups and deduce a description of the free completely simple semigroups, whose existence was proved by McAlister in 1968 and whose structure was first given by Clifford in 1979. From this a description of the lattice of varieties of completely simple semigroups is given in terms of certain subgroups of a free group of countable rank. Whilst not providing a “list” of identities on completely simple semigroups it does enable us to deduce, for instance, the description of all varieties of completely simple semigroups with abelian subgroups given by Rasin in 1979. It also enables us to describe the maximal subgroups of the “free” idempotent-generated completely simple semigroups T(α, β) denned by Eberhart et al. in 1973 and to show in general the maximal subgroups of the “V-free” semigroups of this type (which we define) need not be free in any variety of groups.

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S0308210500020138

Completely simple semigroups: free products, free semigroups and varieties

Embedding semigroups in semibands

Idempotent-generated completely O-simple semigroups

Suppose S is a regular semigroup and E is its set of idempotents. If E is subsemigroup of S, then S has been called orthodox and studied recently by Hall [3], Meakin [6], and Yamada [8]. In this paper we assume that E is not (necessarily) a subsemigroup of S and consider the subsemigroup generated by E, denoted . If E denotes the set of all elements of S which can be written E, denoted . If E denotes the set of all elements of S which can be written as the product of n (not necessarily distinct) idempotents of S, then . We show that is always a regular subsemigroup of S and investigate relationships between it and S. The case where = S is of particular interest to us; such semigroups will be referred to as idempotent-generated regular semi- groups.

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/611F2BE6DBB2A397284DEB729626DEDF/S1446788700012726a.pdf/div-class-title-idempotent-generated-regular-semigroups-div.pdf

W. Williams

Papers

Idempotent-generated regular semigroups