Categorical abstract algebraic logic: Ordered equational logic and algebraizable povarieties
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.
scientific article; zbMATH DE number 5146175
scientific article