B. Plotkin

Universal Algebra, Algebraic Logic, and Databases

Preface. Introduction: 0. General View on Objectives and Contents of the Book. I: Universal Algebra. 1. Sets, Algebras, Models. 2. Fundamental Structures. 3. Categories. 4. The Categories of Sets. Topoi. Fuzzy Sets. 5. Varieties of Algebras. Axiomatizable Classes. 6. Category Algebra and Algebraic Theories. II: Algebraic Logic. 7. Boolean Algebras and Propositional Calculus. 8. Halmos Algebras and Predicate Calculus. 9. Specialized Halmos Algebras. 10. Connections with Model Theory. 11. The Categorial Approach to Algebraic Logic. III: Databases -- Algebraic Aspects. 12. Algebraic Model of a Database. 13. Equivalence and Reorganization of Databases. 14. Symmetries of Relations and Galois Theory of Databases. 15. Constructions in Database Theory. 16. Discussion and Conclusion. Bibliography. Index.

Varieties of algebras and algebraic varieties

The paper contains a brief account of ideas and results, which are described in [1] and [2] with details and proofs. The subject of the paper is algebraic geometry in arbitrary algebraic structures.

Automorphisms of categories of free algebras of varieties

Let Θ be an arbitrary variety of algebras and let Θ0 be the category of all free finitely generated algebras from Θ. We study automorphisms of such categories for special Θ. The cases of the varieties of all groups, all semigroups, all modules over a noetherian ring, all associative and commutative algebras over a field are completely investigated. The cases of associative and Lie algebras are also considered. This topic relates to algebraic geometry in arbitrary variety of algebras Θ.

Geometrical equivalence of groups

The notion of geometrical equivalence of two algebras, which is basic for this paper, is introduced in [5], [6]. It is motivated in the framework of universal algebraic geometry, in which algebraic varieties are considered in arbitrary varieties of algebras. Universal algebraic geometry (as well as classic algebraic geometry) studies systems of equations and its geometric images, i.e., algebraic varieties, consisting of solutions of equations. Geometrical equivalence of algebras means, in some sense, equal possibilities for solving systems of equations. In this paper we consider results about geometrical equivalence of algebras, and special attention is paied on groups (abelian and nilpotent).

Algebra, categories and databases

Publisher Summary This chapter surveys the connections between algebra and databases (DBs). An algebraic model of DBs and applications of this model are emphasized. A database is an information system which allows to store and process information as well as to query its contents. It is possible to query either the data directly, stored in the DB, or some information which can be derived from the basic data. The derived information is expressed in terms of basic data, through some algebraic means. Groups of automorphisms of DBs play a significant role, and are used for solving specific problems such as Galois Theory of DBs, which is used for solving the problem of equivalence of two DBs. The evaluation of the possibilities of algebra in the theory of DBs is provided. The applications of monads and comonads in DBs are also focused. Monads are used for enrichment of DB structure, while comonads work in the model of computations. The three main traditional approaches in DBs are distinguished as: hierarchical; network; and relational approaches. The semantics modeling problems, computational problems, and creation of various DBMSs are considered in the chapter.

ON AUTOMORPHISMS OF CATEGORIES OF UNIVERSAL ALGEBRAS

Let be a variety of universal algebras. We suggest an approach for describing automorphisms of a category of free -algebras. In particular, this approach allows us to answer the question: is an automorphism of such a category inner? Most of the results actually deal with arbitrary categories supplied with a represented forgetful functor.

ALGEBRAIC LOGIC AND LOGICALLY-GEOMETRIC TYPES IN VARIETIES OF ALGEBRAS

The main objective of this paper is to show that the notion of type which was developed within the frames of logic and model theory has deep ties with geometric properties of algebras. These ties go back and forth from universal algebraic geometry to the model theory through the machinery of algebraic logic. We show that types appear naturally as logical kernels in the Galois correspondence between filters in the Halmos algebra of first order formulas with equalities and elementary sets in the corresponding affine space.

Automorphisms of Categories of Free Modules, Free Semimodules, and Free Lie Modules∗

In algebraic geometry over a variety of universal algebras Θ, the group Aut(Θ0) of automorphisms of the category Θ0 of finitely generated free algebras of Θ is of great importance. In this article, semi-inner automorphisms are defined for the categories of free (semi)modules and free Lie modules; then, under natural conditions on a (semi)ring, it is shown that all automorphisms of those categories are semi-inner. We thus prove that for a variety Rℳ of semimodules over an IBN-semiring R (an IBN-semiring is a semiring analog of a ring with IBN), all automorphisms of Aut(Rℳ0) are semi-inner. Therefore, for a wide range of rings, this solves Problem 12 left open in Plotkin (2002); in particular, for Artinian (Noetherian, PI-) rings R, or a division semiring R, all automorphisms of Aut(Rℳ0) are semi-inner.

ON AUTOMORPHISMS OF THE ENDOMORPHISM SEMIGROUP OF A FREE UNIVERSAL ALGEBRA

Let U be a universal algebra. An automorphism α of the endomorphism semigroup of U defined by α(φ) = sφs-1 for a bijection s : U → U is called a quasi-inner automorphism. We characterize bijections on U defining such automorphisms. For this purpose, we introduce the notion of a pre-automorphism of U. In the case when U is a free universal algebra, the pre-automorphisms are precisely the well-known weak automorphisms of U. We also provide different characterizations of quasi-inner automorphisms of endomorphism semigroups of free universal algebras and reveal their structure. We apply obtained results for describing the structure of groups of automorphisms of categories of free universal algebras, isomorphisms between semigroups of endomorphisms of free universal algebras, automorphism groups of endomorphism semigroups of free Lie algebras etc.

Multi-sorted logic and logical geometry: some problems

Abstract The paper has a form of a survey on basics of logical geometry and consists of three parts. It is focused on the relationship between many-sorted theory, which leads to logical geometry and one-sorted theory, which is based on important model-theoretic concepts. Our aim is to show that both approaches go in parallel and there are bridges which allow to transfer results, notions and problems back and forth. Thus, an additional freedom in choosing an approach appears. A list of problems which naturally arise in this field is another objective of the paper.