Categorical abstract algebraic logic: The largest theory system included in a theory family

Abstract

MSC (2000)03G99, 18C15, 08C05, 08B05, 68N30In this note, it is shown that, given a π-institution I = Sign, SEN, C , with N a category of natural transformations on SEN, every theory family T of I includes a unique largest theory system ← T of I. ← T satisfies the important property that its N -Leibniz congruence system always includes that of T . As a consequence, it is shown, on the one hand, that the relation Ω N ( ← T ) = Ω N (T ) characterizes N -protoalgebraicity inside the class of N -prealgebraic π-institutions and, on the other, that all N -Leibniz theory families associated with theory families of a protoalgebraic π-institution I are in fact N -Leibniz theory systems.

Categorical Abstract Algebraic Logic: Ordered Equational Logic and Algebraizable PoVarieties

“…The original motivation of Pałasińska and Pigozzi was the development of a part of the theory G. Voutsadakis (B) School of Mathematics and Computer Science, Lake Superior State University, 650 W. Easterday Avenue, Sault Sainte Marie, MI 49783, USA e-mail: gvoutsad@lssu.edu of abstract algebraic logic (AAL) suitable for handling logical implication in a way analogous to the way logical equivalence is handled by the well-known (Leibniz and Tarski) operator approach in AAL (see, for instance [2,3,6,8,[20][21][22]27] and the surveys [7,16,17]). Since, in recent work by the author (see [32][33][34][35][36][37][38][39][40][41]), algebraic systems were shown to play a role analogous to that of algebras in the study of logical equivalence in the categorical framework, it is only natural to expect that an approach towards logical implication analogous to that adopted by Pałasińska and Pigozzi in [26] will involve the study of partially ordered algebraic systems or partially ordered functors (pofunctors), as introduced and studied in the preceding three papers of this series [42][43][44].…”

“…Finally, a few general references on concepts used in this paper: for background from category theory the reader is referred to any of [1,5,23], for an overview of the current state of affairs in abstract algebraic logic the reader may consult the review article [17], the monograph [16] and the book [7], whereas for more recent developments on the categorical side of the subject the reader may refer to the series of papers [32][33][34][35][36][37][38][39][40][41] in the given order.…”

Categorical Abstract Algebraic Logic: Bloom's Theorem for Rule-Based -Institutions

“…The author has recently further abstracted the theory of abstract algebraic logic [28]- [37] to cover, apart from sentential logics, also logics that are formalized as πinstitutions [14]. In the present work, based on the notion of a category of natural transformations, that has proven key to the development of the theory of categorical abstract algebraic logic, a syntax is developed for π-institutions.…”