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COVERS FOR S-ACTS AND CONDITION (A) FOR A MONOID S

Part of: Semigroups

Published online by Cambridge University Press:  19 December 2014

ALEX BAILEY ,
VICTORIA GOULD ,
MIKLÓS HARTMANN ,
JAMES RENSHAW  and
LUBNA SHAHEEN
ALEX BAILEY
Affiliation:
School of Mathematics, University of Southampton, Highfield Southampton SO17 1BJ, United Kingdom e-mail: alex.bailey@soton.ac.uk
VICTORIA GOULD
Affiliation:
Department of Mathematics, University of York, Heslington, York YO10 5DD, United Kingdom e-mail: victoria.gould@york.ac.uk
MIKLÓS HARTMANN
Affiliation:
Department of Mathematics, University of York, Heslington, York YO10 5DD, United Kingdom e-mail: miklos.hartmann@york.ac.uk
JAMES RENSHAW
Affiliation:
School of Mathematics, University of Southampton, Highfield, Southampton SO17, 1BJ, United Kingdom e-mail: J.H.Renshaw@soton.ac.uk
LUBNA SHAHEEN
Affiliation:
Department of Mathematics, University of York, Heslington, York YO10 5DD, United Kingdom e-mail: lls502@york.ac.uk
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Abstract

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A monoid S satisfies Condition (A) if every locally cyclic left S-act is cyclic. This condition first arose in Isbell's work on left perfect monoids, that is, monoids such that every left S-act has a projective cover. Isbell showed that S is left perfect if and only if every cyclic left S-act has a projective cover and Condition (A) holds. Fountain built on Isbell's work to show that S is left perfect if and only if it satisfies Condition (A) together with the descending chain condition on principal right ideals, MR. We note that a ring is left perfect (with an analogous definition) if and only if it satisfies MR. The appearance of Condition (A) in this context is, therefore, monoid specific. Condition (A) has a number of alternative characterisations, in particular, it is equivalent to the ascending chain condition on cyclic subacts of any left S-act. In spite of this, it remains somewhat esoteric. The first aim of this paper is to investigate the preservation of Condition (A) under basic semigroup-theoretic constructions. Recently, Khosravi, Ershad and Sedaghatjoo have shown that every left S-act has a strongly flat or Condition (P) cover if and only if every cyclic left S-act has such a cover and Condition (A) holds. Here we find a range of classes of S-acts $\mathcal{C}$ such that every left S-act has a cover from $\mathcal{C}$ if and only if every cyclic left S-act does and Condition (A) holds. In doing so we find a further characterisation of Condition (A) purely in terms of the existence of covers of a certain kind. Finally, we make some observations concerning left perfect monoids and investigate a class of monoids close to being left perfect, which we name left$\mathcal{IP}$a-perfect.

MSC classification

Primary: 20M30: Representation of semigroups; actions of semigroups on sets
Secondary: 20M50: Connections of semigroups with homological algebra and category theory
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 57 , Issue 2 , May 2015 , pp. 323 - 341
Copyright
Copyright © Glasgow Mathematical Journal Trust 2014 

References

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