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.

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. \omega^r = \{(e,f) \, :\, fe = e\} \omega^l = \{ (e,f)\, :\, ef = e \} R = \omega^r\, \cap \, (\omega^r)^{-1} L = \omega^l\, \cap \, (\omega^l)^{-1} \omega = \omega^r \, \cap \, \omega^l If T is any statement about E involving the partial binary operation and the above relations in E, one can define the left-right dual of T denoted by T*. If D E is symmetric then T* is meaningful whenever T is. (B31) If and then gf = ( )f. (B32) If and if f and g are in then ( )e = ( )( ). In (the M-set of e and f in that order), define a relation \prec by g \prec h\quad \Longleftrightarrow \quad eg \,\,\omega^r\,\, eh\,,\,\,\, gf\,\, \omega^l \,\,hf . Then the set S(e,f) = \{ h\in M(e,f) : g\prec h \text{ for all } g\in M(e,f) \} is called the sandwich set of e and f in that order. (B4) If f and g are in ω r ( ) then S( )e = S ( ). A mapping φ : E → F between two biordered sets E and F is a biordered set homomorphism (also called a bimorphism) if for all ( ) in D E we have ( ) ( ) = ( )φ. where V = A ⊕ B means that A and B are subspaces of V and V is the internal direct sum of A and B. The partial binary operation ⋆ on E defined by makes E a biordered set. The quasiorders in E are characterised as follows:

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