Affiliated Composition Series of S -acts via B -Clear and B -Identical Subacts

In this paper, we introduce a new type of composition series for [Formula: see text]-acts over a monoid [Formula: see text], built upon the notions of [Formula: see text]-clear and [Formula: see text]-identical acts. These concepts extend the ideas of [Formula: see text]-dense and [Formula: see text]-single acts and provide a framework for composition series affiliated to a fixed act [Formula: see text]. We establish conditions for the existence of such series and describe how their lengths relate to those of subacts and factor acts. Finally, we connect this affiliated theory to classical composition series via the Hom-functor.