Categorical abstract algebraic logic: Ordered equational logic and algebraizable povarieties (Q878147)

This paper contains the fourth (and final) installment on research concerning an extension of some of the results on partially ordered varieties and quasi-varieties of partially ordered universal algebras obtained by Palasińska and Pigozzi in the context of abstract algebraic logic and reported in \textit{D. Pigozzi}'s lecture notes available at \url{http://www.math.iastate.edu/dpigozzi}. A syntactic apparatus is introduced for the study of the algebraic properties of classes of partially ordered algebraic systems (a.k.a. partially ordered functors (pofunctors)). A Birkhoff-style order HSP theorem and a Mal'tsev-style order SLP theorem are proved for partially ordered varieties and partially ordered quasivarieties, respectively, of partially ordered algebraic systems based on this syntactic apparatus. Finally, the notion of a finitely algebraizable partially-ordered quasivariety, in the spirit of Palasińska and Pigozzi, is introduced and some of the properties of these quasi-povarieties are explored in the categorical framework.