In [H. Rashidi, A. Golchin and H. Mohammadzadeh saany, On [Formula: see text]-flat acts, Categ. Gen. Algebr. Struct. Appl. 12(1) (2020) 175–197], the study of [Formula: see text]-flatness property of right acts [Formula: see text] over a monoid [Formula: see text] that can be described by means of when the functor [Formula: see text]-preserves some pullbacks is initiated. In this paper, we extend these results to [Formula: see text]-posets and present equivalent description of [Formula: see text]-po-flatness of [Formula: see text]-posets. We show that [Formula: see text]-flatness does not imply torsion freeness in [Formula: see text]-posets and give some general properties and a characterization of pomonoids for which some other properties of their posets imply this condition.
Yuqun Chen and K. P. Shum in [Rees short exact sequence of S-systems, Semigroup Forum 65 (2002), 141–148] introduced Rees short exact sequence of acts and considered conditions under which a Rees short exact sequence of acts is left and right split, respectively. To our knowledge, conditions under which the induced sequences by functors Hom(RLS, –), Hom(–, RLS) and AS ⊗ S– (where R, S are monoids) are exact, are unknown. This article addresses these conditions. Results are different from that of modules.
In this article, we present GP W-flatness property of acts over monoids, which is a generalization of principal weak flatness. We say that a right S-act AS is GP W-flat if for every s ∈ S, there exists a natural number n = n (s,A S) ∈ N such that the functor AS ⊗ S − preserves the embedding of the principal left ideal S (Ss n) into S S. We show that a right S-act AS is GP W-flat if and only if for every s ∈ S there exists a natural number n = n (s,A S) ∈ N such that the corresponding ϕ is surjective for the pullback diagram P (Ss n , Ss n , ι, ι, S), where ι : S (Ss n) → S S is a monomorphism of left S-acts. Also we give some general properties and a characterization of monoids for which this condition of their acts implies some other properties and vice versa.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.