×

\(E^*\)-unitary inverse semigroups. (English) Zbl 1061.20056

Gomes, Gracinda M. S. (ed.) et al., Semigroups, algorithms, automata and languages. Proceedings of workshops held at the International Centre of Mathematics, CIM, Coimbra, Portugal, May, June and July 2001. Singapore: World Scientific (ISBN 981-238-099-X/hbk). 195-214 (2002).
In the paper with each inverse monoid with zero \(S\) a category \(C(S)\) is associated, which is a slight modification of the category Leech associated with an inverse monoid.
Using this category characterizations of \(E^*\)-unitary, strongly \(E^*\)-unitary and \(E\)-unitary inverse monoids are given. Specifically, it is proved that: 1) \(S\) is \(E^*\)-unitary if and only if \(C(S)\) is cancellative; 2) \(S\) is strongly \(E^*\)-unitary if and only if \(C(S)\) can be embedded in a groupoid; 3) \(S\) is \(E\)-unitary (with a zero adjoined) if and only if \(C(S)\) has a groupoid of fractions.
For an inverse semigroup a “universal group” is introduced and it is shown how this group can be used to characterize strongly \(E^*\)-unitary semigroups.
For the entire collection see [Zbl 1005.00031].

MSC:

20M18 Inverse semigroups
20M50 Connections of semigroups with homological algebra and category theory
18B40 Groupoids, semigroupoids, semigroups, groups (viewed as categories)
PDFBibTeX XMLCite