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
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
editReferences
edit- ^ a b c d e f 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
- ^ Mitsch, H. (2009). "Subdirect products of E–inversive semigroups". Journal of the Australian Mathematical Society. 48: 66–78. doi:10.1017/S1446788700035199.
- ^ 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
- ^ 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): 233–248. doi:10.1007/s002330010131.
- ^ Fountain, J.; Hayes, A. (2014). "E ∗-dense E-semigroups". Semigroup Forum. 89: 105–124. doi:10.1007/s00233-013-9562-z. preprint
Further reading
edit- 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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