Adjoint Relations Between the Category of Poset Acts and Some Other Categories

Abstract

In this paper, first the congruences in the category PosAct-S of all poset acts over a pomonoid S; an S-act in the category Pos of all posets, with action preserving monotone maps between them, are introduced. Then, we study the existence of the free and cofree objects in the category PosAct-S. More precisely, we consider all forgetful functors between this category and the categories Pos-S of all S-posets, Pos of all posets, Act-S of all S-acts, and Set of all sets, and we study the existence of their left and right adjoints. It is shown that the category Pos-S is a full reflective and coreflective subcategory of PosAct-S.

Injective and projective poset acts

“…The category of poset acts with action preserving monotone maps between them, first has been introduced and studied by Skornyakov in [19], [20] and continued by Shahbaz in [16], where congruences in this category are introduced and characterized and adjoint relations between this category and the categories Pos of posets, Act-S of S-acts, and Set of sets, are discussed.…”