Categorical Abstract Algebraic Logic Research Articles

Categorical Abstract Algebraic Logic: Wojcicki's Conjecture and Malinowski's Theorem

Categorical Abstract Algebraic Logic: Wojcicki's Conjecture and Malinowski's Theorem

Categorical abstract algebraic logic: skywatching in semilattice systems

Font and Moraschini established a bijective correspondence between congruences of semilattices with sectionally finite height and certain special subsets of their universes, called clouds. They provided a characterization of clouds and showed that the correspondence is given by the Leibniz operator of abstract algebraic logic. We extend the bijection to one between congruence systems on the semilattice systems of categorical abstract algebraic logic and what we call cloud families. In this context, the categorical analog of the Leibniz operator plays a similar role. In addition, we show that, even though the exact analogue of the Font–Moraschini condition fails in general, a more complex variant provides an analogous characterization of cloud families.

Categorical Abstract Algebraic Logic: Truth-Equational π-Institutions

Categorical Abstract Algebraic Logic: Truth-Equational π-Institutions

Categorical Abstract Algebraic Logic: Referential π-Institutions

Wojcicki introduced in the late 1970s the concept of a referential semantics for propositional logics. Referential semantics incorporate features of the Kripke possible world semantics for modal logics into the realm of algebraic and matrix semantics of arbitrary sentential logics. A well-known theorem of Wojcicki asserts that a logic has a referential semantics if and only if it is selfextensional. Referential semantics was subsequently studied in detail by Malinowski and the concept of selfextensionality has played, more recently, an important role in the field of abstract algebraic logic in connection with the operator approach to algebraizability. We introduce and review some of the basic definitions and results pertaining to the referential semantics of π-institutions, abstracting corresponding results from the realm of propositional logics.

Categorical Abstract Algebraic Logic: Referential Algebraic Semantics

Wojcicki has provided a characterization of selfextensional logics as those that can be endowed with a complete local referential semantics. His result was extended by Jansana and Palmigiano, who developed a duality between the category of reduced congruential atlases and that of reduced referential algebras over a fixed similarity type. This duality restricts to one between reduced atlas models and reduced referential algebra models of selfextensional logics. In this paper referential algebraic systems and congruential atlas systems are introduced, which abstract referential algebras and congruential atlases, respectively. This enables the formulation of an analog of Wojcicki's Theorem for logics formalized as ?-institutions. Moreover, the results of Jansana and Palmigiano are generalized to obtain a duality between congruential atlas systems and referential algebraic systems over a fixed categorical algebraic signature. In future work, the duality obtained in this paper will be used to obtain one between atlas system models and referential algebraic system models of an arbitrary selfextensional ?-institution. Using this latter duality, the characterization of fully selfextensional deductive systems among the selfextensional ones, that was obtained by Jansana and Palmigiano, can be extended to a similar characterization of fully selfextensional ?-institutions among appropriately chosen classes of selfextensional ones.

Categorical Abstract Algebraic Logic: Meet-Combination of Logical Systems

The widespread and rapid proliferation of logical systems in several areas of computer science has led to a resurgence of interest in various methods for combining logical systems and in investigations into the properties inherited by the resulting combinations. One of the oldest such methods isfibring. In fibring the shared connectives of the combined logics inherit properties frombothcomponent logical systems, and this leads often to inconsistencies. To deal with such undesired effects, Sernadas et al. (2011, 2012) have recently introduced a novel way of combining logics, calledmeet-combination, in which the combined connectives share only thecommonlogical properties they enjoy in the component systems. In their investigations they provide a sound and concretely complete calculus for the meet-combination based on available sound and complete calculi for the component systems. In this work, an effort is made to abstract those results to a categorical level amenable tocategorical abstract algebraic logictechniques.

Categorical Abstract Algebraic Logic: Structurality, protoalgebraicity, and correspondence

AbstractThe notion of an ℐ ‐matrix as a model of a given π ‐institution ℐ is introduced. The main difference from the approach followed so far in Categorical Abstract Algebraic Logic (CAAL) and the one adopted here is that an ℐ ‐matrix is considered modulo the entire class of morphisms from the underlying N ‐algebraic system of ℐ into its own underlying algebraic system, rather than modulo a single fixed (N,N ′)‐logical morphism. The motivation for introducing ℐ ‐matrices comes from a desire to formulate a correspondence property for N ‐protoalgebraic π ‐institutions closer in spirit to the one for sentential logics than that considered in CAAL before. As a result, in the previously established hierarchy of syntactically protoalgebraic π ‐institutions, i. e., those with an implication system, and of protoalgebraic π ‐institutions, i. e., those with a monotone Leibniz operator, the present paper interjects the class of those π ‐institutions with the correspondence property, as applied to ℐ ‐matrices. Moreover, this work on ℐ ‐matrices enables us to prove many results pertaining to the local deduction‐detachment theorems, paralleling classical results in Abstract Algebraic Logic formulated, first, by Czelakowski and Blok and Pigozzi. Those results will appear in a sequel to this paper. (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Categorical Abstract Algebraic Logic: Local Characterization Theorems for Classes of Systems

Let ℒ = ⟨F, R, ρ⟩ be a system language. Given a class of ℒ-systems K and an ℒ-algebraic system A = ⟨SEN,⟨N,F⟩⟩, i.e., a functor SEN: Sign → Set, with N a category of natural transformations on SEN, and F:F → N a surjective functor preserving all projections, define the collection K A of A-systems in K as the collection of all members of K of the form 𝔄 = ⟨ SEN,⟨N,F⟩,R 𝔄 ⟩, for some set of relation systems R 𝔄 on SEN. Taking after work of Czelakowski and Elgueta in the context of the model theory of equality-free first-order logic, several relationships between closure properties of the class K, on the one hand, and local properties of K A and global properties connecting K A and K A′, whenever there exists an ℒ-morphism ⟨ F,α⟩ : A → A′, on the other, are investigated. In the main result of the article, it is shown, roughly speaking, that K A is an algebraic closure system, for every ℒ-algebraic system A, provided that K is closed under subsystems and reduced products.

Categorical Abstract Algebraic Logic: Equivalential π-Institutions

The theory of equivalential deductive systems, as introduced by Prucnal and Wroński and further developed by Czelakowski, is abstracted to cover the case of logical systems formalized as π-Institutions. More precisely, the notion of an N-equivalence system for a given π-Institutions is introduced. A characterization theorem for N-equivalence systems, previously proven for N-parameterized equivalence systems, is revisited and a “transfer theorem” for N-equivalence systems is proven. For a π-Institutions I having an N-equivalence system, the maximum such system is singled out and, then, an analog of Herrmann’s Test, characterizing those N-protoalgebraic π-Institutions having an N-equivalence system, is formulated. Finally, some of the rudiments of matrix theory are revisited in the context of π-Institutions, as they relate to the existence of N-equivalence systems.

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

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

Categorical abstract algebraic logic: The Diagram and the Reduction Operator Lemmas

AbstractThe study of structure systems, an abstraction of the concept of first‐order structures, is continued. Structure systems have algebraic systems as their algebraic reducts and their relational component consists of a collection of relation systems on the underlying functors. An analog of the expansion of a first‐order structure by constants is presented. Furthermore, analogs of the Diagram Lemma and the Reduction Operator Lemma from the theory of equality‐free first‐order structures are provided in the framework of structure systems. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Categorical Abstract Algebraic Logic: Prealgebraicity and Protoalgebraicity

Two classes of π are studied whose properties are similar to those of the protoalgebraic deductive systems of Blok and Pigozzi. The first is the class of N-protoalgebraic π-institutions and the second is the wider class of N-prealgebraic π-institutions. Several characterizations are provided. For instance, N-prealgebraic π-institutions are exactly those π-institutions that satisfy monotonicity of the N-Leibniz operator on theory systems and N-protoalgebraic π-institutions those that satisfy monotonicity of the N-Leibniz operator on theory families. Analogs of the correspondence property of Blok and Pigozzi for π-institutions are also introduced and their connections with preand protoalgebraicity are explored. Finally, relations of these two classes with the (\({\mathcal{I}}\), N)-algebraic systems, introduced previously by the author as an analog of the \({\mathcal{S}}\) -algebras of Font and Jansana, and with an analog of the Suszko operator of Czelakowski for π-institutions are also investigated.

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

Categorical Abstract Algebraic Logic: Subdirect Representation of Pofunctors

Pałasińska and Pigozzi developed a theory of partially ordered varieties and quasi-varieties of algebras with the goal of addressing issues pertaining to the theory of algebraizability of logics involving an abstract form of the connective of logical implication. Following their lead, the author has abstracted the theory to cover the case of algebraic systems, systems that replace algebras in the theory of categorical abstract algebraic logic. In this note, an order subdirect representation theorem for partially ordered algebraic systems is proven. This is an analog of the Order Subdirect Representation Theorem of Pałasińska and Pigozzi, which, in turn, generalizes the well-known Subdirect Representation Theorem of Universal Algebra.

Categorical Abstract Algebraic Logic: Leibniz Equality and Homomorphism Theorems

The study of structure systems, an abstraction of the concept of first-order 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 structure systems are introduced and the Leibniz congruence system of a structure system is defined. Analogs of the Homomorphism, the Second Isomorphism and the Correspondence Theorems of Universal Algebra are provided in this more abstract context. These results generalize corresponding results of Elgueta for equality-free first-order logic. Finally, a version of Godel’s Completeness Theorem is provided with reference to structure systems.

Categorical 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 as an analog of protoalgebraic sentential logics with the goal of extending the Leibniz hierarchy from the sentential framework to the π-institution framework. Many properties of protoalgebraic logics, studied in the sentential logic framework by Blok and Pigozzi, Czelakowski, and Font and Jansana, among others, have already been adapted in previous work by the author to the categorical level. This work aims at further advancing that study by exploring in this new level some more properties of protoalgebraic sentential logics.

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

AbstractIn 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 $ \overleftarrow T $ of ℐ. $ \overleftarrow 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 ΩN ($ \overleftarrow T $) = ΩN (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. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Categorical Abstract Algebraic Logic: Partially Ordered Algebraic Systems

An extension of parts of the theory of partially ordered varieties and quasivarieties, as presented by Palasinska and Pigozzi in the framework of abstract algebraic logic, is developed in the more abstract framework of categorical abstract algebraic logic. Algebraic systems, as introduced in previous work by the author, play in this more abstract framework the role that universal algebras play in the more traditional treatment. The aim here is to build the generalized framework and to formulate and prove abstract versions of the ordered homomorphism theorems in this framework.

Corrigendum to “Categorical abstract algebraic logic: The criterion for deductive equivalence”

We give a correction to the paper [2] mentioned in the title. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

Categorical Abstract Algebraic Logic: Models of π-Institutions

An important part of the theory of algebraizable sentential logics consists of studying the algebraic semantics of these logics. As developed by Czelakowski, Blok, and Pigozzi and Font and Jansana, among others, it includes studying the properties of logical matrices serving as models of deductive systems and the properties of abstract logics serving as models of sentential logics. The present paper contributes to the development of the categorical theory by abstracting some of these model theoretic aspects and results from the level of sentential logics to the level of π-institutions.