Akbar Golchin

Subpullbacks and Flatness Properties of S-Posets

In [5, pp. 829–850] study was initiated of flatness properties of right acts A S over a monoid S that can be described in terms when the functor A S ⊗-preserves pullbacks. In that article, familiar flatness properties emerged in a new light, and new properties such as (PWP) and (WP) were discovered. In this article, we extend these results to S-posets. Also we introduce Conditions (WP) w and (PWP) w and consider the relation between them and Conditions (WP), (PWP), and po-flatness.

STRONGLY TORSION FREE ACTS OVER MONOIDS

An act AS is called torsion free if for any a, b ∈ AS and for any right cancellable element c ∈ S the equality ac = bc implies a = b. In [M. Satyanarayana, Quasi- and weakly-injective S-system, Math. Nachr.71 (1976) 183–190], torsion freeness is considered in a much stronger sense which we call in this paper strong torsion freeness and will characterize monoids by this property of their (cyclic, monocyclic, Rees factor) acts.

On Properties of Product Acts over Monoids

Let S be a monoid. It is known that flatness may be defined in a number of ways for the class of right S − acts, and these notions themselves naturally give rise to some weaker associated conditions. To date no work has been done on these properties of product acts. This article is concerned with closure under products of the class of right S − acts possessing one of these “flatness” properties and conversely, when these properties transfer from products to components. We consider such questions for both a general monoid S, and for monoids coming from some special classes. Specifically, we answer the question of when properties such as torsion freeness, principal weak flatness, GP − flatness, (weak) flatness, Conditions (P), (P′), etc. transfer from products of acts to their components. Also we give equivalences of when properties such as torsion freeness, Conditions (PWP), (P′) (EP), etc. transfer from acts to their products. Finally, we extend some results from Bulman-Fleming, S., Gilmour, A.

Characterization of monoids by condition (GP)

In this paper, we introduce a generalization of Condition (P), called Condition (GP), and will characterize monoids by this condition of their right (Rees factor) acts. Furthermore, we will show that Conditions (GP) and (E) are interpolation type conditions for strong flatness.

On deleted diagonal acts of completely (0-) simple semigroups

If S is a semigroup without identity, then the deleted diagonal act of S is the right S1-act Dd(S) := (S × S)S1. In [S. Bulman-Fleming and A. Gilmour, Flatness properties of diagonal acts over monoids, Semigroup Forum 79 (2009) 298–314] the authors answered the question of when Dd(S) is flat, satisfies Condition (P) or (E) for a completely (0-) simple semigroup (always represented here in regular Rees matrix form). In this paper we answer similar question for some other properties. There are also some results that can arise.