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

By Alex Bailey, Victoria Gould, Miklos Hartmann, James Renshaw and Lubna Shaheen


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.<br/><br/>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.<br/><br/>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.<br/><br/>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

Topics: QA75
Year: 2015
OAI identifier: oai:eprints.soton.ac.uk:347140
Provided by: e-Prints Soton

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  3. (2000). Monoids, Acts, and Categories, de Gruyter, doi
  4. (2008). On covers of cyclic acts over monoids’, doi
  5. (2012). On flatness covers of cyclic acts over monoids’, doi
  6. (2000). On flatness properties of cyclic acts’, doi
  7. (1996). On monoids over which all strongly flat cyclic right acts are projective’, doi
  8. (1971). Perfect monoids’, doi
  9. (1976). Perfect semigroups’, doi
  10. (2010). Perfection for pomonoids’, doi
  11. (1972). Projectivity of acts and Morita equivalence of monoids’, doi
  12. (2010). Strongly flat and condition (P) covers of acts over monoids’, doi

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