## Covers for S-acts and Condition (A) for a monoid S

Research output: Contribution to journalArticlepeer-review

### Standard

Covers for S-acts and Condition (A) for a monoid S. / Bailey, Alex; Gould, Victoria; Hartmann, Miklos; Renshaw, James; Shaheen, Lubna.

In: Glasgow Mathematical Journal, Vol. 57, No. 2, 01.05.2015, p. 323-341.

Research output: Contribution to journalArticlepeer-review

### Harvard

Bailey, A, Gould, V, Hartmann, M, Renshaw, J & Shaheen, L 2015, 'Covers for S-acts and Condition (A) for a monoid S', Glasgow Mathematical Journal, vol. 57, no. 2, pp. 323-341. https://doi.org/10.1017/S0017089514000317

### APA

Bailey, A., Gould, V., Hartmann, M., Renshaw, J., & Shaheen, L. (2015). Covers for S-acts and Condition (A) for a monoid S. Glasgow Mathematical Journal, 57(2), 323-341. https://doi.org/10.1017/S0017089514000317

### Vancouver

Bailey A, Gould V, Hartmann M, Renshaw J, Shaheen L. Covers for S-acts and Condition (A) for a monoid S. Glasgow Mathematical Journal. 2015 May 1;57(2):323-341. https://doi.org/10.1017/S0017089514000317

### Author

Bailey, Alex ; Gould, Victoria ; Hartmann, Miklos ; Renshaw, James ; Shaheen, Lubna. / Covers for S-acts and Condition (A) for a monoid S. In: Glasgow Mathematical Journal. 2015 ; Vol. 57, No. 2. pp. 323-341.

@article{cdda3fc41f52465e8719c0423d9c605d,
title = "Covers for S-acts and Condition (A) for a monoid S",
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 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 such that every left S-act has a cover from 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 lefta-perfect.",
keywords = "S-acts, projective, strongly flat, Condition (A), cover",
author = "Alex Bailey and Victoria Gould and Miklos Hartmann and James Renshaw and Lubna Shaheen",
note = "This is an author-produced version of the published paper. Uploaded in accordance with the publisher{\textquoteright}s self-archiving policy. Further copying may not be permitted; contact the publisher for details ",
year = "2015",
month = may,
day = "1",
doi = "10.1017/S0017089514000317",
language = "English",
volume = "57",
pages = "323--341",
journal = "Glasgow Mathematical Journal",
issn = "1469-509X",
publisher = "Cambridge University Press",
number = "2",

}

TY - JOUR

T1 - Covers for S-acts and Condition (A) for a monoid S

AU - Bailey, Alex

AU - Gould, Victoria

AU - Hartmann, Miklos

AU - Renshaw, James

AU - Shaheen, Lubna

N1 - This is an author-produced version of the published paper. Uploaded in accordance with the publisher’s self-archiving policy. Further copying may not be permitted; contact the publisher for details

PY - 2015/5/1

Y1 - 2015/5/1

N2 - 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 such that every left S-act has a cover from 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 lefta-perfect.

AB - 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 such that every left S-act has a cover from 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 lefta-perfect.

KW - S-acts, projective, strongly flat, Condition (A), cover

U2 - 10.1017/S0017089514000317

DO - 10.1017/S0017089514000317

M3 - Article

VL - 57

SP - 323

EP - 341

JO - Glasgow Mathematical Journal

JF - Glasgow Mathematical Journal

SN - 1469-509X

IS - 2

ER -