Dual of Extending Acts

Since 1980s, the study of the extending module in the module theory has been a major area of research interest in the ring theory and it has been studied recently by several authors, among them N.V. Dung, D.V. Huyn, P.F. Smith and R. Wisbauer. Because the act theory signifies a generalization of the module theory, the author studied in 2017 the class of extending acts which are referred to as a generalization of quasi-injective acts. The importance of the extending acts motivated us to study a dual of this concept, named the coextending act. An S-act MS is referred to as coextending act if every coclosed subact of Ms is a retract of MS where a subact AS of MS is said to be coclosed in MS if whenever the Rees factor ⁄ is small in the Rees factor ⁄then AS=BS for each subact BS of AS. Various properties of this class of acts have been examined. Characterization of this concept is intended to show the behavior of a coextending property. In addition, based on the results obtained by us, the conditions under which subacts inherit a coextending property were demonstrated. Ultimately, a part of this paper

Characterization of monoids by condition (GP)

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

Finite semigroups that are minimal for not being Malcev nilpotent

We give a description of finite semigroups S that are minimal for not being Malcev nilpotent, i.e. every proper subsemigroup and every proper Rees factor semigroup is Malcev nilpotent but S is not. For groups this question was considered by Schmidt.

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.

The ideal structure of nilpotent-generated transformation semigroups

In 1987 Sullivan determined the elements of the semigroup N(X) generated by all nilpotent partial transformations of an infinite set X; and later in 1997 he studied subsemigroups of N(X) defined by restricting the index of the nilpotents and the cardinality of the set. Here, we describe the ideals and Green's relations on such semigroups, like Reynolds and Sullivan did in 1985 for the semigroup generated by all idempotent total transformations of X. We then use this information to describe the congruences on certain Rees factor semigroups and to construct families of congruence-free semigroups with interesting algebraic properties. We also study analogous questions for X finite and for one-to-one partial transformations.

A congruence-free semigroup associated with an infinite cardinal number

SynopsisLet X be a set with infinite cardinality m and let Qm be the semigroupof balanced elements in ℐ(X), as described by Howie. If I is the ideal{αεQm:|Xα|<m} then the Rees factor Pm = Qm/I is O-bisimple and idempotent-generated. Its minimum non-trivial homomorphic image has both these properties and is congruence-free. Moreover, has depth 4, in the sense that [E()]4 = , [E()]3≠