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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.<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

- Covers of acts over monoids and pure epimorphisms’, to appear in
- Monoids for which Condition (P) acts are projective’, Semigroup Forum 61 (2000), 46–56. E-mail address: alex.bailey@soton.ac.uk School of Mathematics,
- (2000). Monoids, Acts, and Categories, de Gruyter,
- (2008). On covers of cyclic acts over monoids’,
- (2012). On ﬂatness covers of cyclic acts over monoids’,
- (2000). On ﬂatness properties of cyclic acts’,
- (1996). On monoids over which all strongly ﬂat cyclic right acts are projective’,
- (1971). Perfect monoids’,
- (1976). Perfect semigroups’,
- (2010). Perfection for pomonoids’,
- (1972). Projectivity of acts and Morita equivalence of monoids’,
- (2010). Strongly ﬂat and condition (P) covers of acts over monoids’,

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