Biordered set
A biordered set (otherwise known as boset) is a mathematical object that occurs in the description of the structure of the set of idempotents in a semigroup. The set of idempotents in a semigroup is a biordered set and every biordered set is the set of idempotents of some semigroup. A regular biordered set is a biordered set with an additional property. The set of idempotents in a regular semigroup is a regular biordered set, and every regular biordered set is the set of idempotents of some regular semigroup. History The concept and the terminology were developed by K S S Nambooripad in the early 1970s. In 2002, Patrick Jordan introduced the term boset as an abbreviation of biordered set. The defining properties of a biordered set are expressed in terms of two quasiorders defined on the set and hence the name biordered set. According to Mohan S. Putcha, "The axioms defining a biordered set are quite complicated. However, considering the general nature of semigroups, it is rather surprising that such a finite axiomatization is even possible." Since the publication of the original definition of the biordered set by Nambooripad, several variations in the definition have been proposed. David Easdown