Perfection for pomonoids

A pomonoid S is a monoid equipped with a partial order that is compatible with the binary operation. In the same way that M -acts over a monoid M correspond to the representation of M by transformations of sets, S -posets correspond to the representation of a pomonoid S by order preserving transformations of posets. Following standard terminology from the theories of R -modules over a unital ring R , and M -acts over a monoid M , we say that a pomonoid S is left poperfect if every left S -poset has a projective cover. Left perfect rings were introduced in 1960 in a seminal paper of Bass (Trans. Am. Math. Soc. 95, 446–488, 1960 ) and shown to be precisely those rings satisfying M R , the descending chain condition on principal right ideals. In 1971, inspired by the results of Bass and Chase (Trans. Am. Math. Soc. 97, 457–473, 1960 ), Isbell was the first to study left perfect monoids Isbell (Semigroup Forum 2, 95–118, 1971 ). The results of Isbell, together with those of Fountain (Proc. Edinb. Math. Soc. 20, 87–93, 1976 ), show that a monoid is left perfect if and only if it satisfies a finitary condition dubbed Condition (A), in addition to M R . Moreover, M R can be replaced by another finitary condition, namely Condition (D). A further characterisation of left perfect rings was given in Chase ( 1960 ), where Chase proved that a ring is left perfect if and only if every flat left module is projective; the corresponding result for M -acts was demonstrated in Fountain ( 1976 ). In this paper we continue the study of left poperfect pomonoids, recently initiated in Pervukhin and Stepanova (Algebra Logic 48, 54–71, 2009 ). We show, as in Pervukhin and Stepanova, that a pomonoid S is left poperfect if and only if it satisfies ( M R ) and the ‘ordered’ version Condition (A o ) of Condition (A) and further, these conditions are equivalent to every strongly flat left S -poset being projective. On the other hand, we argue via an analysis of direct limits that Conditions (A) and (A o ) are equivalent, so that a pomonoid S is left perfect if and only if it is left poperfect. We also give a characterisation of left poperfect monoids involving the ordered version of Condition (D). Our results and many of our techniques certainly correspond to those for monoids, but we must take careful account of the partial ordering on S , and in places introduce alternative strategies to those found in Isbell, Fountain and Pervukhin and Stepanova.