Biordered sets and fundamental semigroups

Abstract

Given any biordered set E, a natural construction yields a semigroup T E that is always fundamental, in the sense that T E possesses no nontrivial idempotent-separating congruence. In the case that E=E(S) is the biordered set of idempotents of a semigroup S generated by regular elements, there is a natural representation of S by T E , such that S becomes a biorder-preserving coextension of a fundamental and symmetric subsemigroup of T E . If further S is regular then this yields the fundamental constructions of Nambooripad, Grillet and Hall, which in turn generalise the construction of Munn of a maximum fundamental inverse semigroup from its semilattice of idempotents.