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.{{cite book|editor=Gracinda M. S. Gomes|title=Semigroups, Algorithms, Automata and Languages|chapter-url=https://books.google.com/books?id=IL58mAsfXOgC&pg=PA167|year=2002|publisher=World Scientific|isbn=978-981-277-688-4|pages=167–168|author=John Fountain|chapter=An introduction to covers for semigrops}} [http://www-users.york.ac.uk/~jbf1/coimbra2.pdf preprint] 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.{{Cite journal | doi = 10.1017/S1446788700035199| title = Subdirect products of E–inversive semigroups| journal = Journal of the Australian Mathematical Society| volume = 48| pages = 66–78| year = 2009| last1 = Mitsch | first1 = H.| doi-access = free}}Manoj Siripitukdet and Supavinee Sattayaporn [http://www.sci.nu.ac.th/rs/upload/s/2550010001nation19.pdf Semilattice Congruences on E-inversive Semigroups] {{Webarchive|url=https://web.archive.org/web/20140903063942/http://www.sci.nu.ac.th/rs/upload/s/2550010001nation19.pdf |date=2014-09-03 }}, NU Science Journal 2007; 4(S1): 40 - 44G. Thierrin (1955), 'Demigroupes inverses et rectangularies', Bull. Cl. Sci. Acad. Roy. Belgique 41, 83-92. Some authors use E-dense to refer only to E-inversive semigroups in which the idempotents commute.{{Cite journal | doi = 10.1007/s002330010131| title = Certain congruences on E-inversive E-semigroups| journal = Semigroup Forum| volume = 65| issue = 2| pages = 233–248| year = 2002| last1 = Weipoltshammer | first1 = B. }}

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.{{Cite journal | doi = 10.1007/s00233-013-9562-z| title = E ∗-dense E-semigroups| journal = Semigroup Forum| volume = 89| pages = 105–124| year = 2014| last1 = Fountain | first1 = J. | last2 = Hayes | first2 = A. }} [http://maths.york.ac.uk/www/sites/default/files/Preprint_No%2006_2013_2014.pdf preprint]