__NOTOC__ In

abstract algebra In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras over a field. The term ''a ...

, 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 se ...

''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 (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 Abrahamic religions (Judaism, Christianity and Islam), Gabriel (); Greek: grc, Γαβριήλ, translit=Gabriḗl, label=none; Latin: ''Gabriel''; Coptic: cop, Ⲅⲁⲃⲣⲓⲏⲗ, translit=Gabriêl, label=none; Amharic: am, ገብ� ...

in 1955. Some authors use ''E''-dense to refer only to ''E''-inversive semigroups in which the idempotents commute. More generally, a

subsemigroup 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'', ...

''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 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 ...

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 regular semigroup is ''E''-dense (but not vice versa). * Any

eventually regular semigroup In mathematics, a semigroup is a nonempty set together with an associative binary operation. A special class of semigroups is a class of semigroups satisfying additional properties or conditions. Thus the class of commutative semigroups consis ...

is ''E''-dense. * Any

periodic semigroup In mathematics, a monogenic semigroup is a semigroup generated by a single element. Monogenic semigroups are also called cyclic semigroups. Structure The monogenic semigroup generated by the singleton set is denoted by \langle a \rangle . The s ...

(and in particular, any

finite semigroup Finite is the opposite of infinite. It may refer to: * Finite number (disambiguation) * Finite set, a set whose cardinality (number of elements) is some natural number * Finite verb, a verb form that has a subject, usually being inflected or marked ...

) is ''E''-dense.

Examples

See also

References

Further reading