In this paper, we establish the necessary and sufficient condition for an equivalence relation ρ on S-act A endowed with a topology such that A/ρ becomes Hausdorff topological S-act. Also, show if A1 A2 be two S-acts, then any homomorphism ϕ : → A2, A1/ ker is only ϕ-saturated continuous. Moreover, congruences θ1 θ2 topology, ∩ congruence mapping A/θ1 × A/θ2, defined by ϕ(a) = (aθ1, aθ2), all ∈...

in this paper $s$ is a monoid with a left zero and $a_s$ (or $a$) is a unitary right $s$-act. it is shown that a monoid $s$ is right perfect (semiperfect) if and only if every (finitely generated) strongly flat right $s$-act is quasi-projective. also it is shown that if every right $s$-act has a unique zero element, then the existence of a quasi-projective cover for each right act implies that ...

B efore meiotic divisions, chromosomes form crossovers, which represent the reciprocal exchanges between the DNA molecules of the homolo-gous partner chromosomes. These crossovers aren't just important for reassorting genetic traits; they're essential for the correct alignment and segregation of meiotic chromosomes. To ensure that every homologous chromosome pair forms at least one cross-over—a...

In this paper $S$ is a monoid with a left zero and $A_S$ (or $A$) is a unitary right $S$-act. It is shown that a monoid $S$ is right perfect (semiperfect) if and only if every (finitely generated) strongly flat right $S$-act is quasi-projective. Also it is shown that if every right $S$-act has a unique zero element, then the existence of a quasi-projective cover for each right act implies that ...

In this paper, we show that injectivity with respect to the class $mathcal{D}$  of dense monomorphisms of an idempotent and weakly hereditary closure operator of an arbitrary category  well-behaves. Indeed, if $mathcal{M}$ is a subclass of monomorphisms, $mathcal{M}cap mathcal{D}$-injectivity  well-behaves. We also introduce the notion of $(r,t)$-injectivity in the category {bf S-Act}, where $r...