On Characterization of Monoids by Properties of Generators

Abstract:

      Kilp and Knauer in (Comm. Algebra, 1992, 20(7), 1841--1856) gave characterizations of monoids when all generators in category of right $S$-acts ($S$ is a monoid) satisfy properties such as freeness, projectivity, strong flatness, Condition (P), principal weak flatness, principal weak injectivity, weak injectivity, injectivity, divisibility, strong faithfulness and torsion freeness. Sedaghtjoo in (Semigroup Forum, 2013, 87: 653--662) characterized monoids by some other properties of generators including weak flatness, Condition (E) and regularity. To our knowledge, the problem has not been studied for properties mentioned above of (finitely generated, cyclic, monocyclic, Rees factor) right acts. In this article we answer the question corresponding to these properties and also $fg$-weak injectivity.