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.[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 aS there exists another element bS such that ab is an idempotent.
  • for every element aS there exists another element cS 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 xS, there exists yS such that both xyT and yxT.

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]


See also[edit]


  1. ^ 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
  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, 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[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