CATEGORICAL ABSTRACT ALGEBRAIC LOGIC: OPERATORS ON CLASSES OF STRUCTURE SYSTEMS
George Voutsadakis · Scientiae mathematicae Japonicae · 2006
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. A variety of operators on classes of structure systems are introduced and studied, taking after similar work of Elgueta in the context of the model theory of equality-free first-order logic. Both Elgueta's and the present work are inspired by considerations arising in the study of the process of algebraization in abstract algebraic logic. The ways that these various class operators interact, when composed with one-another, are at the focus of current investigations. 1 Introduction The central role that classes of logical matrices play in the theory of abstract algebraic logic (see, e.g., (8)) together with the fact that logical matrices may be viewed as models of universal Horn logic without equality (3) (see also (8)) provided the motivation for the study of the model theory of equality-free first-order structures by Dellunde, Elgueta and their collaborators (see (9, 13, 14, 15, 16, 17) for Elgueta's work, some of which is joint with Czelakowski and some with Jansana, and (6, 10, 11, 12) for Dellunde's work, some of which is joint with Casanovas and Jansana). The idea was that equality-free first-order model theory, which, as contrasted with its counterpart with equality, was not as well studied, may benefit from results inspired by its interaction with abstract algebraic logic and that, conversely, some novel ideas in that theory may prove useful in the domain of the algebraization of sentential logics and the theory of logical matrices. In recent work by the author (23, 24, 25) the theory of algebraizability of sentential logics has been abstracted to cover those logical systems that are formalized as π-institutions. The class of all these systems is wider than that of sentential logics since it includes logics with multiple signatures and quantifiers. Moreover, the π-institution presentation is, in some ways, more attractive from the metalogical point of view since it allows the treatment of substitutions in the object language rather than delegating them to the metalanguage. On the other hand, one has to pay the price that the added generality restricts, to a certain extent, both the quantity and the depth of the results obtained, since these apply now to a wider variety of logical systems. Ongoing investigations, however, allow optimism that the amount of results that one is still able to obtain is worth the effort and, also, that some of these results may prove fruitful in reconsidering or adding to the existing knowledge pertaining to the theory as applied specifically to sentential logics. The concept of a logical matrix, when lifted to the π-institution framework, gives rise to that of a matrix system (26). The concept of an abstract logic, which was used extensively by Font and Jansana in (18) as an alternative algebraic model for sentential logics, more