Varieties of complex algebras

This paper is an introduction: in particular, to algebras of relations of various ranks, and in general, to the part of algebraic logic algebraizing quantifier logics. The paper has a survey character, too. The most frequently used algebras like cylindric-, relation-, polyadic-, and quasi-polyadic algebras are carefully introduced and intuitively explained for the nonspecialist. Their variants, connections with logic, abstract model theory, and further algebraic logics are also reviewed. Efforts were made to make the review part relatively comprehensive. In some directions we tried to give an overview of the most recent results and research trends, too.

Algebraization of quantifier logics, an introductory overview

The modal logic of provability. The sequential approach

Dominical categories are categories in which the notions of partial morphisms and their domains become explicit, with the latter being endomorphisms rather than subobjects of their sources. These categories form the basis for a novel abstract formulation of recursion theory, to which the present paper is devoted. The abstractness has of course its usual concomitant advantage of generality: it is interesting to see that many of the fundamental results of recursion theory remain valid in contexts far removed from their classic manifestations. A principal reason for introducing this new formulation is to achieve an algebraization of the generalized incompleteness theorem, by providing a category-theoretic development of the concepts and tools of elementary recursion theory that are inherent in demonstrating the theorem. Dominical recursion theory avoids the commitment to sets and partial functions which is characteristic of other formulations, and thus allows for an intrinsic recursion theory within such structures as polyadic algebras. It is worthy of notice that much of elementary recursion theory can be developed without reference to elements . By Godel's generalized incompleteness theorem for consistent arithmetical system T we mean any statement of the following sort: (1) if every recursive set is definable in T , then T is essentially undecidable [41]; or (2) if all recursive functions are definable in T , then T is essentially undecidable [41]; or (3) if every recursive set is definable in T , then T 0 and R 0 (the sets of Godel numbers of the theorems and refutables of T ) are recursively inseparable [39]; or (4) if all re sets are representable in T , then T 0 is creative [28], [39]; or (5) if T is a Rosser theory (i.e., all disjoint re sets are strongly separable in T ), then T 0 and R 0 are effectively inseparable [39].

Dominical categories: Recursion theory without elements

Elementary generation and canonicity for varieties of Boolean algebras with operators

Algebra rarely has anything deep to say about logic. Most applications of algebra to logic are fairly shallow, the exceptions being applications of representation theory, where one is really using the non-algebraic properties of the representations. Nonetheless, algebra does offer another perspective and a convenient language or framework in which to work. The purpose of the present chapter is the presentation of such an algebraic framework-language in which to place the results already discussed, in which goals like those of the last chapter can be expressly delineated, and in which we can state and prove a theorem that delimits, not very convincingly, the boundaries for such successful generalisations of Solovay’s Completeness Theorems as were obtained in the last chapter and explains, again not convincingly, the necessarily close relation between these results and Solovay’s results for PRL.

C. Smoryński

Papers

Fixed point algebras