Coherent Monoids

Abstract

Let S be a monoid. A (unital) right S-set A is called (principally) weakly flat if the functor A ® -preserves embeddings of (principal) left ideals into S, and flat if this functor preserves all embeddings of left S-sets. S is called (weakly) (left, right) absolutely flat if all (left, right) S-sets are (weakly) flat. Our paper [3] contains results which will be used in the sequel, and is also suggested as a general reference on flatness.It is easily seen that coproducts of families of (weakly) flat right S-sets are again (weakly) flat. Taking our lead from ring theory, we shall call S (weakly) left coherent if all direct products of non-empty families of (weakly) flat right S-sets are (weakly) flat. (Weak) right coherence is defined similarly and S is called (weakly) coherent if it is both (weakly) left and right coherent. In [5] S. U. Chase proved that the following conditions (among others) are equivalent, if R is a ring with identity:(1) R r is a flat right R-module, for any non-empty set r.(2) Every finitely generated left ideal of R is finitely presented.(3) (a) For every aER the left annihilator (0: a) = {rER I ra = O} is a finitely generated left ideal, and (b) the intersection of any two finitely generated left ideals of R is finitely generated. (4) The direct product of any family of flat right R-modules is again flat.2Research supported by Natural Sciences and Engineering Research Council of Canada grants A4494 and A9241.

Coherent Monoids

“…Thus if a monoid S is weakly right coherent in the sense of [4] then it is weakly right coherent in our sense; this also follows indirectly from Corollary 3.3 and Theorem 4 of [4]. COROLLARY …”

“…We remark that in view of the comments following Proposition 5.4, parts (iii), (iv) and (v) of Corollary 3.6 also follow from [4,Theorem 7].…”

“…One can of course ask the corresponding question for products and it transpires that the answers are related. On the other hand we recall that in [4] a monoid S is defined to be (weakly) right coherent if every non-empty product of (weakly) flat left 5-sets is (weakly) flat. We now consider a further notion of coherency, which appears naturally when considering weakly flat ultraproducts of left 5-sets and is moreover related to the work in [4].…”

“…The final section considers some relations between weak right coherency, products and ultraproducts of flat left 5-sets, and the work of BulmanFleming and McDowell [4]. In view of Los's theorem this is also connected with the question of determining, for which monoids S is the class of flat left 5-sets (first order) axiomatisable?…”

On direct products of S-posets satisfying flatness properties

“…Gould in [6] then solved this problem for strongly flat and conditions (P) and (E). Meanwhile, Bulman-Fleming and McDowell [4] defined a monoid to be right coherent if every direct product of flat S -acts is flat, and they obtained some results when S Γ is (principally) weakly flat for a monoid S . In [2], Bulman-Fleming and Gilmour discussed when S × S has certain flatness properties.…”