Narrow Results

The object of this paper is to generalize the notion of supplement in modules to monoid acts. In contrast to the case of modules that supplements of submodules do not generally exist, here we uniquely characterize the supplement of a proper subact of an act. Supplemented acts are defined as acts whose proper subacts all have proper supplements. We discuss how the property of being supplemented relates to certain other properties of acts. In particular, we prove that being supplemented and being completely reducible coincide.

This study is concerned with the collections of prime and radical subacts of free acts over a semigroup $S.$ The collection of all prime subacts of any free act is topologized with two topologies as generalizations of the notion of the Zariski topology in rings and semigroups. Moreover, the interplay between some algebraic properties and some topological properties of the spectrum of free acts is investigated. In particular, the conditions under which the set of all prime subacts of free acts is a spectral space are studied. Also, it is shown that a certain type of Zariski space of free acts is isomorphic to a lattice of radical subacts.

An act A_S is called torsion free if for any a, b ∈ A_S 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.

In 2006, Condition $(E^{\prime }P)$ was introduced by Golchin and Mohammadzadeh and the theorems associated with it were proved. In this paper, we continue the study in relation to the above condition and for some of the properties in the $S$-acts. We present the equivalent on the basis of that, which in turn illustrates the importance of this condition. Given that Condition $(E^{\prime }P)$ is the generalization of Conditions $(P^{\prime })$ and $(E^{\prime })$, so the study of the condition in relation to other properties of the $S$-acts will give a better understanding of the behavior of the $S$-acts and even Conditions $(P^{\prime })$ and $(E^{\prime })$.

In this paper, we define measurable $S$-acts on monoids and construct a new category, namely Meas Act-$S$. We investigate the behavior of this category with respect to product, coproduct, pushout, pullback, equalizer and coequalizer. Next, we study the injectivity and complete injectivity in category Meas Act-$S$ and show any measurable $S$-act can be embedded into complete injectivity. Moreover, we investigate the Skornjakov’ theorem for measurable S-acts.

Let L_S denote the language of (right) S-acts over a monoid S and let Σ_S be a set of sentences in L_S which axiomatises S-acts. A general result of model theory says that Σ_S has a model companion, denoted by T_S, precisely when the class of existentially closed S-acts is axiomatisable and in this case, T_S axiomatises . It is known that T_S exists if and only if S is right coherent. Moreover, by a result of Ivanov, T_S has the model-theoretic property of being stable.

In the study of stable first order theories, superstable and totally transcendental theories are of particular interest. These concepts depend upon the notion of type: we describe types over T_S algebraically, thus reducing our examination of T_S to consideration of the lattice of right congruences of S. We indicate how to use our result to confirm that T_S is stable and to prove another result of Ivanov, namely that T_S is superstable if and only if S satisfies the maximal condition for right ideals. The situation for total transcendence is more complicated but again we can use our description of types to ascertain for which right coherent monoids S we have that T_S is totally transcendental and is such that the U-rank of any type coincides with its Morley rank.