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Covers for S-acts and condition (A) for a monoid S

Covers for S-acts and condition (A) for a monoid S
Covers for S-acts and condition (A) for a monoid S
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 article 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 C such that every left S-act has a cover from 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 IPa-perfect.
0017-0895
323-341
Bailey, Alex
cd2762de-6a67-4ffc-ab42-2842ca378fa8
Gould, Victoria
4f27cdf3-02d1-448d-b2dd-ba5efdf5cff0
Hartmann, Miklos
c0a3bd94-65e5-42c6-8689-046e3a68d926
Renshaw, James
350100c1-f7c7-44d3-acfb-29b94f21731c
Shaheen, Lubna
031f6ead-ea23-4b09-a95f-bd8dd71d8565
Bailey, Alex
cd2762de-6a67-4ffc-ab42-2842ca378fa8
Gould, Victoria
4f27cdf3-02d1-448d-b2dd-ba5efdf5cff0
Hartmann, Miklos
c0a3bd94-65e5-42c6-8689-046e3a68d926
Renshaw, James
350100c1-f7c7-44d3-acfb-29b94f21731c
Shaheen, Lubna
031f6ead-ea23-4b09-a95f-bd8dd71d8565

Bailey, Alex, Gould, Victoria, Hartmann, Miklos, Renshaw, James and Shaheen, Lubna (2015) Covers for S-acts and condition (A) for a monoid S. Glasgow Mathematical Journal, 57 (2), 323-341.

Record type: Article

Abstract

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 article 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 C such that every left S-act has a cover from 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 IPa-perfect.

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Submitted date: 17 January 2013
Accepted/In Press date: 26 January 2014
Published date: May 2015
Organisations: Pure Mathematics

Identifiers

Local EPrints ID: 347140
URI: https://eprints.soton.ac.uk/id/eprint/347140
ISSN: 0017-0895
PURE UUID: 7a78d6c0-1d0d-427e-b979-8c72af1144bf

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Date deposited: 17 Jan 2013 12:15
Last modified: 08 Nov 2017 01:31

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Contributors

Author: Alex Bailey
Author: Victoria Gould
Author: Miklos Hartmann
Author: James Renshaw
Author: Lubna Shaheen

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