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.[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
  • 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]
gollark: Last month I got a xenowyrm invisiprize.
gollark: But, you know, invisibly.
gollark: My invisiprize this month cycles through the rainbow in its animation.
gollark: Aah. Network errors.
gollark: To the no killing.

See also

References
  1. John Fountain (2002). "An introduction to covers for semigrops". In Gracinda M. S. Gomes (ed.). Semigroups, Algorithms, Automata and Languages. World Scientific. pp. 167–168. ISBN 978-981-277-688-4. preprint
  2. Mitsch, H. (2009). "Subdirect products of E–inversive semigroups". Journal of the Australian Mathematical Society. 48: 66. doi:10.1017/S1446788700035199.
  3. Manoj Siripitukdet and Supavinee Sattayaporn Semilattice Congruences on E-inversive Semigroups Archived 2014-09-03 at the Wayback Machine, NU Science Journal 2007; 4(S1): 40 - 44
  4. G. Thierrin (1955), 'Demigroupes inverses et rectangularies', Bull. Cl. Sci. Acad. Roy. Belgique 41, 83-92.
  5. Weipoltshammer, B. (2002). "Certain congruences on E-inversive E-semigroups". Semigroup Forum. 65 (2): 233. doi:10.1007/s002330010131.
  6. Fountain, J.; Hayes, A. (2014). "E ∗-dense E-semigroups". Semigroup Forum. 89: 105. doi:10.1007/s00233-013-9562-z. preprint

Further reading
  • 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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