regular semigroup

To see this, let S be a regular semigroup in which idempotents commute.

There are two equivalent ways in which to define a regular semigroup S:

"The natural partial order on a regular semigroup".

The name "completely regular semigroup" stems from Lyapin's book on semigroups.

In a regular semigroup S, every - and -class contains at least one idempotent.

The homomorphic image of a regular semigroup is regular.

In general Nambooripad's order in a regular semigroup is not compatible with multiplication.

The partial order in a regular semigroup discovered by Nambooripad can be defined in several equivalent ways.

Alternatively, a regular semigroup is inverse if and only if any two idempotents commute.

A regular semigroup is a semigroup in which every element is regular.

In a completely regular semigroup, each Green H-class is a group and the semigroup is the union of these groups.

Definitions (regular semigroup)

Conversely, it can be shown that any inverse semigroup is a regular semigroup in which idempotents commute.

Orthodox semigroups: a regular semigroup S is orthodox if its subset of idempotents forms a subsemigroup.

A regular semigroup is an inverse semigroup if and only if it admits an involution under which each idempotent is an invariant.

Also, if a regular semigroup S has a p-system that is multiplicatively closed (i.e. subsemigroup), then S is an inverse semigroup.

Locally inverse semigroups: a regular semigroup S is locally inverse if eSe is an inverse semigroup, for each idempotent e.

Every element of S has at least one inverse (S is a regular semigroup) and idempotents commute (that is, the idempotents of S form a semilattice).

The idempotents of B appear down the diagonal, in accordance with the fact that in a regular semigroup with commuting idempotents, each L-class and each R-class must contain exactly one idempotent.

In mathematics, Nambooripad order (also called Nambooripad's partial order) is a certain natural partial order on a regular semigroup discovered by K S S Nambooripad in late seventies.

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.

Let S be any regular semigroup and S be the semigroup obtained by adjoining the identity 1 to S. For any x in S let R be the Green R-class of S containing x.