Categorical Abstract Algebraic Logic: Subdirect Representation of Pofunctors

Pałasińska and Pigozzi developed a theory of partially ordered varieties and quasi-varieties of algebras with the goal of addressing issues pertaining to the theory of algebraizability of logics involving an abstract form of the connective of logical implication. Following their lead, the author has abstracted the theory to cover the case of algebraic systems, systems that replace algebras in the theory of categorical abstract algebraic logic. In this note, an order subdirect representation theorem for partially ordered algebraic systems is proven. This is an analog of the Order Subdirect Representation Theorem of Pałasińska and Pigozzi, which, in turn, generalizes the well-known Subdirect Representation Theorem of Universal Algebra.