In this paper, we first define the notion of $\mathcal{F}$-cosmall quotient for an additive exact substructure $\mathcal{F}$ structure $\mathcal{E}$ in category $\mathcal{A}$. We show that every is right minimal some cases. also give definition $\mathcal{F}$-superfluous and relate it approximation modules. As application, investigate our results a pure-exact $\mathcal{F}$.

Weakly recognizing morphisms from free semigroups onto finite semigroups are a classical way for defining the class of ω-regular languages, i.e., a set of infinite words is weakly recognizable by such a morphism if and only if it is accepted by some Büchi automaton. We consider the descriptional complexity of various constructions for weakly recognizing morphisms. This includes the conversion f...