Categorical Abstract Algebraic Logic: Compatibility Operators and the Leibniz Hierarchy
نویسنده
چکیده
A unified treatment of the operator approach to categorical abstract algebraic logic (CAAL) was recently presented by the author using as tools the notions of compatibility operator of Czelakowski, of coherent compatibility operator of Albuquerque, Font and Jansana and exploiting an abstract Galois connection established via the use of these operators. The approach encompasses previous work by the author, but it also enriches the semantic, i.e., operator-based, side of the categorical Leibniz hierarchy with many new results. In this paper, we continue the work by providing, inter alia, characterizations of the categorical analogs of the classes of the Leibniz hierarchy based on full generalized matrix systems and on various properties of the categorical Leibniz and Suszko operators. School of Mathematics and Computer Science, Lake Superior State University, Sault Sainte Marie, MI 49783, USA, [email protected]
منابع مشابه
Categorical Abstract Algebraic Logic: Compatibility Operators and Correspondence Theorems
Very recently Albuquerque, Font and Jansana, based on preceding work of Czelakowski on compatibility operators, introduced coherent compatibility operators and used Galois connections, formed by these operators, to provide a unified framework for the study of the Leibniz, the Suszko and the Tarski operators of abstract algebraic logic. Based on this work, we present a unified treatment of the o...
متن کاملCategorical Abstract Algebraic Logic: More on Protoalgebraicity
Protoalgebraic logics are characterized by the monotonicity of the Leibniz operator on their theory lattices and are at the lower end of the Leibniz hierarchy of abstract algebraic logic. They have been shown to be the most primitive among those logics with a strong enough algebraic character to be amenable to algebraic study techniques. Protoalgebraic π -institutions were introduced recently a...
متن کاملCategorical Abstract Algebraic Logic: Leibniz Equality and Homomorphism Theorems
The study of structure systems, an abstraction of the concept of firstorder structures, is continued. Structure systems have algebraic systems rather than universal algebras as their algebraic reducts. Moreover, their relational component consists of a collection of relation systems on the underlying functors rather than simply a system of relations on a single set. Congruence systems of struct...
متن کاملAssertional logics, truth-equational logics, and the hierarchies of abstract algebraic logic
We establish some relations between the class of truth-equational logics, the class of assertional logics, other classes in the Leibniz hierarchy, and the classes in the Frege hierarchy. We argue that the class of assertional logics belongs properly in the Leibniz hierarchy. We give two new characterizations of truth-equational logics in terms of their full generalized models, and use them to o...
متن کاملLogics of varieties, logics of semilattices and conjunction
This paper starts with a general analysis of the problem of how to associate a logic with a given variety of algebras, and shows that it has a positive solution for two of the standard procedures of performing this association, and a negative one in the third. Then the paper focuses on the case of the “logics of semilattices”, which are defined as the logics related to the variety of semilatice...
متن کامل