Most of the techniques of combining logics are devoted to produce new logics from given ones. This is the case, for instance, of fibring and fusion. The other side of the coin, that is, how to decompose a logic into others ('splitting' a logic, as opposite to 'splicing' logics) has been less explored in the literature. Possible-Translations Semantics (PTS's), introduced by W. Carnielli in the 90's, are one of the main representatives of this approach, and several applications of PTS's were obtained, including a generalization of Blok-Pigozzi's methods for algebraizing logics. In this talk we propose a wide notion of morphism between propositional signatures by using multifunctions, and prove that the category of signatures and such morphisms has products. This category is used as a basis for a generalization of PTS's called Possible-Translations Coverings (PTC's). The possibility of obtaining algebraizability of logics via PTC's will be also discussed.