E-dense semigroup

In abstract algebra, an E-dense semigroup is a semigroup in which every element a'' has at least one weak inverse x, meaning that xax = x. The notion of weak inverse is weaker than the notion of inverse used in a regular semigroup.
The above definition of an E-inversive semigroup S is equivalent with any of the following:
for every element a ∈ S there exists another element b ∈ S such that ab is an idempotent.
for every element a ∈ S there exists another element c ∈ S such that ca is an idempotent.
This explains the name of the notion as the set of idempotents of a semigroup S is typically denoted by E.
The concept of E-inversive semigroup was introduced by Gabriel Thierrin in 1955. Some authors use E-dense to refer only to E-inversive semigroups in which the idempotents commute.
More generally, a subsemigroup T of S is said dense in S if, for all x ∈ S, there exists y ∈ S such that both xy ∈ T and yx ∈ T.
A semigroup with zero is said to be an E*-dense semigroup if every element other than the zero has at least one non-zero weak inverse. Semigroups in this class have also been called '0-inversive semigroups.'''

Examples

Any regular semigroup is E-dense.
Any eventually regular semigroup is E-dense.
Any periodic semigroup is E-dense.