E-solid regular semigroups and solid binary algebras
Raymond Broeksteeg · Bulletin of the Australian Mathematical Society · 1995
The idempotents of a regular semigroups play an important role in determining the semigroup's structure, and so have been studied by many authors.In [7,8] Nambooripad defined a regular biordered set to be a nonempty set together with a •partial binary operation defined on it that satisfied certain conditions.He proceeded to show that regular biordered sets characterise the idempotents of regular semigroups.For the special case of a locally inverse regular semigroup, Nambooripad [9] found that one may abstractly characterise its set of idempotents as a binary algebra which satisfies a given finite list of identities (where a binary algebra is a nonempty set together with a binary operation defined on it).We call this type of binary algebra a local semilattice algebra; other names used elsewhere in the literature include 'partially associative pseudosemilattice', 'local semilattice', and 'pseudosemilattice'.The main purpose of this thesis is to derive a similar characterisation of the idempotents of E-solid regular semigroups (which will coincide with the usual band of idempotents if the semigroup is orthodox).This characterisation will be called a solid binary algebra.We also establish a concept of variety for regular biordered sets which is analogous to the concept of existence variety (e-variety) introduced by Hall [5].Chapter 1 contains preliminary results on semigroups, biordered sets, and evarieties.In Chapter 2, we give the promised characterisation of the set of idempotents of an E-solid regular semigroup.We first define a solid binary algebra to be a binary algebra which satisfies six given identities, and their duals.For an E-solid regular semigroup 5, we define a binary operation * on E(S) by (Ve, / G E(S)) e * / = the identity element of the group H? f .It is shown that (E(S), *) is a solid binary algebra, which we call the solid binary algebra of S. Conversely, we show that each solid binary algebra is the solid binary algebra of some E-solid regular semigroup.