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HALL'S CONDITION AND IDEMPOTENT RANK OF IDEALS OF ENDOMORPHISM MONOIDS

Published online by Cambridge University Press:  04 February 2008

R. Gray
R. Gray
Affiliation:
School of Mathematics, University of Leeds, Leeds LS2 9JT, UK ([email protected])
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Abstract

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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.

Keywords

Primary 20M20Secondary 05C99matchinggenerating setendomorphism monoididempotent
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 51 , Issue 1 , February 2008 , pp. 57 - 72
Copyright
Copyright © Edinburgh Mathematical Society 2008