E-dense semigroup

In abstract algebra, an E"-dense semigroup (also called an E"-inversive 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 (as the name suggests) weaker than the notion of inverse used in a regular semigroup (which requires that "axa"="a").
The above definition of an "E"-inversive semigroup "S" is equivalent with any of the following:
This explains the name of the notion as the set of idempotents of a semigroup "S" is typically denoted by "E"("S").
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.