# E-dense semigroup

In abstract algebra, an ** E-dense semigroup** (also called an

**) is a semigroup in which every element**

*E*-inversive semigroup*a*has at least one

**weak inverse**

*x*, meaning that

*xax*=

*x*.

^{[1]}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:^{[1]}

- 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*).^{[1]}

The concept of *E*-inversive semigroup was introduced by Gabriel Thierrin in 1955.^{[2]}^{[3]}^{[4]} Some authors use *E*-dense to refer only to *E*-inversive semigroups in which the idempotents commute.^{[5]}

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.**

^{[6]}

## Examples[edit]

- Any regular semigroup is
*E*-dense (but not vice versa).^{[1]} - Any eventually regular semigroup is
*E*-dense.^{[1]} - Any periodic semigroup (and in particular, any finite semigroup) is
*E*-dense.^{[1]}

## See also[edit]

## References[edit]

- ^
^{a}^{b}^{c}^{d}^{e}^{f}John Fountain (2002). "An introduction to covers for semigrops". In Gracinda M. S. Gomes.*Semigroups, Algorithms, Automata and Languages*. World Scientific. pp. 167–168. ISBN 978-981-277-688-4. preprint **^**Mitsch, H. (2009). "Subdirect products of E–inversive semigroups".*Journal of the Australian Mathematical Society*.**48**: 66. doi:10.1017/S1446788700035199.**^**Manoj Siripitukdet and Supavinee Sattayaporn Semilattice Congruences on E-inversive Semigroups, NU Science Journal 2007; 4(S1): 40 - 44**^**G. Thierrin (1955), 'Demigroupes inverses et rectangularies', Bull. Cl. Sci. Acad. Roy. Belgique 41, 83-92.**^**Weipoltshammer, B. (2002). "Certain congruences on E-inversive E-semigroups".*Semigroup Forum*.**65**(2): 233. doi:10.1007/s002330010131.**^**Fountain, J.; Hayes, A. (2014). "E ∗-dense E-semigroups".*Semigroup Forum*.**89**: 105. doi:10.1007/s00233-013-9562-z. preprint

## Further reading[edit]

- Mitsch, H. "Introduction to E-inversive semigroups." Semigroups : proceedings of the international conference ; Braga, Portugal, June 18–23, 1999. World Scientific, Singapore. 2000. ISBN 9810243928

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