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:
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(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.
Examples Any regular semigroup is E-dense (but not vice versa). Any eventually regular semigroup is E-dense. Any periodic semigroup (and in particular, any finite semigroup) is E-dense.
See also Dense set E-semigroup
References
Further reading Mitsch, H. "Introduction to E-inversive semigroups." Semigroups (Braga, 1999), 114–135. World Scientific Publishing Co., Inc., River Edge, NJ, 2000. ISBN 9810243928