Categorical abstract algebraic logic
In (1) * the theory of algebraizable deductive systems was developed. A deductive system S over a language L is said to be algebraizable if there exists a quasi-variety K, over the same language L, and translations from the sentences of the system into equations and vice-versa that, roughly speaking, simulate the deduction over the system in the equational deduction over the quasi-variety and vice-versa and are inverses of each other. In (3) this notion of algebraizability was shown to be a special case of the, so-called, equivalence of deductive systems. It is, in fact, the equivalence of S with the very special 2-deductive system (2) associated with the quasi-variety K;One of the main limitations of this framework is the way in which it handles logics with varying signatures, like equational or first-order logic. Before they can be algebraized in this framework, they must be transformed in a rather artificial way to propositional-like structural counterparts;In this thesis, following (4), the [pi]-institution framework (5, 6, 7) is used to extend the theory of algebraizability to logics with varying signatures. The notion of equivalence for [pi]-institutions is introduced. A characterization of equivalence for a special class of [pi]-institutions, the, so-called, term [pi]-institutions, is obtained along the lines of (2), by exploiting the relation between their categories of theories. The notion of an algebraic institution is then defined. Algebraic institutions roughly correspond to equational 2-deductive systems, but the formalism here is categorical rather than universal algebraic so that it can handle the added generality appropriately. These special institutions are used for the algebraization of arbitrary [pi]-institutions. The example of the equational institution illustrates the general theory. Limiting the scope to a subclass of term [pi]-institutions, the, so-called, theory institutions, and requiring the syntax to remain invariant during the algebraization process, an intrinsic characterization is obtained for algebraizability in the spirit of (1), via the introduction of a generalized Leibniz operator. The study of the invariance of some metalogical properties under equivalence of [pi]-institutions and the detailed development of the algebraic theory used in the algebraization of the equational institution conclude the thesis. ftn*Please refer to dissertation for references.