__NOTOC__ In

abstract algebra In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics), rings, field (mathematics), fields, module (mathe ...

, an ''E''-dense semigroup (also called an ''E''-inversive semigroup) is a

semigroup In mathematics, a semigroup is an algebraic structure consisting of a set together with an associative internal binary operation on it. The binary operation of a semigroup is most often denoted multiplicatively: ''x''·''y'', or simply ''xy'', ...

in which every element ''a'' has at least one

weak inverse In mathematics, the term weak inverse is used with several meanings. Theory of semigroups In the theory of semigroups, a weak inverse of an element ''x'' in a semigroup is an element ''y'' such that . If every element has a weak inverse, the s ...

''x'', meaning that ''xax'' = ''x''.preprint
/ref> The notion of weak inverse is (as the name suggests) weaker than the notion of inverse used in a

regular semigroup In mathematics, a regular semigroup is a semigroup ''S'' in which every element is regular, i.e., for each element ''a'' in ''S'' there exists an element ''x'' in ''S'' such that . Regular semigroups are one of the most-studied classes of 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 Idempotence (, ) is the property of certain operations in mathematics and computer science whereby they can be applied multiple times without changing the result beyond the initial application. The concept of idempotence arises in a number of pl ...

. * 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.preprint
/ref>

Examples

* Any

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.

Examples

See also

References

Further reading