Flatness properties of monocyclic acts

Abstract

Abstract. In a previous paper the authors studied flatness properties of cyclic acts S/p (S denotes a monoid, and p is a right congruence on S), and determined conditions on S under which all flat or weakly flat acts of this type are actually strongly flat or projective. In the present paper attention is restricted to monocyclic acts (cyclic acts in which p is generated by a single pair of elements of S), and further results on such collapsing of flatness properties are obtained.An observation which is used extensively in this study is the fact that for w and t in S with wt ~ t, S/p(w~, t) is fiat if and only if t is a regular element of S.

Pullbacks and Flatness Properties of Acts. Ii

“…It follows that u ∈ S β , and hence ux = x. Finally, from the description of ρ given in (10) we know that x m u = x n+1 for some nonnegative integers m, n. Multiplication on the right by x produces x m+1 = x n+2 , and so we have…”

Flatness properties of acts over commutative, cancellative monoids

“…(3) If s = t then S/p(s, t) is in fact free, so satisfies condition (P). If J is a unit, then p(s, t) = p(\,s^t), and it is well-known (see [4]) that, in this case, S/p(s, t) also satisfies condition (P). On the other hand, if we are given that S/p(s, t) satisfies (P), then by Proposition 2.3, us=vt for some u, veS such that u p(s, t) 1 p(s, t) v. If u = 1 = v then s = t follows, whereas if at least one of u, v is different from 1, then at least one of s, t is necessarily a unit.…”

“…For reasons of brevity, we will define only those terms to be used directly in this paper. For more complete information, we refer the reader to [4] and its bibliography, or to the survey article [1].…”

On congruence compact monoids

“…In any monoid, the right congruence p(wt, t) generated by a pair of the form (wt, t) is easily seen to be contained in the right congruence p(w, 1), which consists of all elements (a, b) such that w m a = w"b for some non-negative integers m and n. (A full characterization of the relation p(wt, t) is given in Lemma 2.2 of [2].) Thus, in the present situation, there exist non-negative integers m k , n k for each keN such that Suppose that s has infinite order.…”