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E-dense semigroup

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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

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: 66–78. 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): 233–248. doi:10.1007/s002330010131.
  6. Fountain, J.; Hayes, A. (2014). "E ∗-dense E-semigroups". Semigroup Forum. 89: 105–124. doi:10.1007/s00233-013-9562-z. preprint

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


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