Categorical Abstract Algebraic Logic: Ordered Equational Logic and Algebraizable PoVarieties

Abstract

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

Categorical Abstract Algebraic Logic: More on Protoalgebraicity

“…The institution L N (I) will be called the N -pseudo-algebraic counterpart or the NLindenbaum-Tarski counterpart of the π-institution I. Algebraic institutions were introduced in [27]. The term pseudo-algebraic counterpart for the institution L N (I) is chosen because of its similarity with algebraic institutions.…”

“…The term pseudo-algebraic counterpart for the institution L N (I) is chosen because of its similarity with algebraic institutions. However, it should be noted that L N (I) may not be genuinely algebraic, since the category Sign is an arbitrary category and not necessarily a full subcategory of a Kleisli category of some nontrivial algebraic theory (one of the requirements for an algebraic institution in [27]). …”

“…The introduction of protoalgebraic π-institutions was a culmination of a long-term project undertaken by the author with the goal of adapting general definitions, methods, and results widely applied to the theory of algebraizability of deductive systems to π-institutions. The first phase of this project, more syntactic in nature and influenced mainly by the works of Blok and Pigozzi ( [2], [4]), appeared in [26] (see also [27], [28], [29]). A second phase, with a model-theoretic flavor, based on the introduction of a categorical Tarski operator and heavily influenced by the work of Font and Jansana and other members of the Barcelona algebraic logic group (see [13] for an overview and an extensive bibliography), led to the series of papers [21], [22], [23], [24], [25], [30], [31], which are precursors to the current work.…”

“…The most significant among those are an analog of the characterization of protoalgebraicity in terms of the form of the full models of the logic (Theorem 3.4 of [13]) and an analog of the isomorphism between the lattice of all filters that generate full models and all AlgS-congruences (Proposition 3.5 of [13]). Finally, in Section 6, an attempt is made to reconcile the model-theoretic operator approach with the more syntactically flavored, translation-based approach that had been the focus of earlier work ( [26], [27], [28]). Given a π-institution I = Sign, SEN, C with N a category of natural transformations on SEN, an algebraiclike counterpart L N (I), called the N -Lindenbaum-Tarski counterpart of I, is associated with I.…”

On the Algebraization of Many-Sorted Logics

“…In this front, our ultimate aim is to understand the relationship between orthomodular lattices as used in the Birkhoff and Von Neumann tradition of quantum logic and the algebraic counterpart of exogenous quantum logic [15]. An important open question is whether and how our approach can be integrated with the work on the algebraization of logics as institutions reported in [20]. Another interesting line of future work is to study the impact of our proposal with respect to the way a logic is represented within many-sorted equational logic in the context of logic combination, namely in the lines of [17].…”