Categorical Abstract Algebraic Logic: Ordered Equational Logic and Algebraizable PoVarieties

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’cev-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 quasi-variety, in the spirit of Pałasińska and Pigozzi, is introduced and some of the properties of these quasi-povarieties are explored in the categorical framework.

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