In mathematics, an orthodox semigroup is a regular semigroup whose set of idempotents forms a subsemigroup. In more recent terminology, an orthodox semigroup is a regular E-semigroup. The term orthodox semigroup was coined by T. E. Hall and presented in a paper published in 1969. Certain special classes of orthodox semigroups have been studied earlier. For example, semigroups which are also unions of groups, in which the sets of idempotents form subsemigroups were studied by P. H. H. Fantham in 1960.
Contents
Examples
Some elementary properties
The set of idempotents in an orthodox semigroup has several interesting properties. Let S be a regular semigroup and for any a in S let V(a) denote the set of inverses of a. Then the following are equivalent:
Structure
The structure of orthodox semigroups have been determined in terms of bands and inverse semigroups. The Hall–Yamada pullback theorem describes this construction. The construction requires the concepts of pullbacks (in the category of semigroups) and Nambooripad representation of a fundamental regular semigroup.