Hall's condition and idempotent rank of ideals and endomorphism monoids

Gray, R (2008) Hall's condition and idempotent rank of ideals and endomorphism monoids. Proceedings of the Edinburgh Mathematical Society, 51 (01). pp. 57-72. ISSN 0013-0915

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In 1990, Howie and McFadden showed that every proper two-sided ideal of the full transformation monoid $T_n$, the set of all maps from an $n$-set to itself under composition, has a generating set, consisting of idempotents, that is no larger than any other generating set. This fact is a direct consequence of the same property holding in an associated finite $0$-simple semigroup. We show a correspondence between finite $0$-simple semigroups that have this property and bipartite graphs that satisfy a condition that is similar to, but slightly stronger than, Hall's condition. The results are applied in order to recover the above result for the full transformation monoid and to prove the analogous result for the proper two-sided ideals of the monoid of endomorphisms of a finite vector space.

Item Type: Article
Faculty \ School: Faculty of Science > School of Mathematics
UEA Research Groups: Faculty of Science > Research Groups > Algebra and Combinatorics
Depositing User: Users 2731 not found.
Date Deposited: 21 Feb 2013 22:59
Last Modified: 16 May 2023 00:32
URI: https://ueaeprints.uea.ac.uk/id/eprint/41525
DOI: 10.1017/S0013091504001397

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