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**covers_for_sacts_and_condition_a_for_a_monoid_s**172 KB, PDF document

Journal | Glasgow Mathematical Journal |
---|---|

Date | Accepted/In press - 14 Jan 2014 |

Date | E-pub ahead of print - 19 Dec 2014 |

Date | Published (current) - 1 May 2015 |

Issue number | 2 |

Volume | 57 |

Number of pages | 19 |

Pages (from-to) | 323-341 |

Early online date | 19/12/14 |

Original language | English |

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.

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

## Representations of Semigroups

Project: Research project (funded) › Research

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