Introduction
   General View on Objectives and contents of the book

UNIVERSAL ALGEBRA

SETS, ALGEBRAS, AND MODELS
       Sets
	Sets, subsets and mappings
	Multiplication of mappings
	Cartesian product of sets
	Free sum of sets
	Characteristic function of sets
	Binary relations
	Equivalences
	Quotient sets
	Fundamental mapping theorem
	Cardinality of a set
	Fuzzy sets

      Algebras and models
	Algebraic operations and relations
     	Algebras
	Models and algebraic systems
	Homomorphism theorem
	Subalgebras and submodels
	Cartesian products
	Remak theorem
	Algebra of terms
	Generators and defining relations
	Classes of algebras and models; their axioms

    Many-sorted systems
	Set complexes
	Many-sorted operations and relations
	Many-sorted algebras and models
	Algebras of many-sorted terms

FUNDAMENTAL STRUCTURES

   Definitions and examples
	Semigroups
	Groups
	Origins of groups and semigroups
	Quasigroups and loops
	Rings
	Fields and skew-fields
	More examples of rings and fields. The origins
	Linear spaces and modules
	Associative linear algebras
	Group algebras and semigroup algebras
	Other structures

   Homomorphisms. Free Algebras.
	Definition of a free algebra
	Semigroups
	Groups
	Rings
	Linear spaces and modules
	Linear algebras

   Some many-sorted structures
	Representations of groups and semigroups
	Linear representations
	Automata
	Affine spaces and affine automata   	
		
CATEGORIES
     General information and examples
	Definition of a category
	Examples
	Subcategories
	Monomorphisms, epimorphisms and isomorphisms
	Duality
	Functors
	Natural transformations of functors. Categories of functors
	Equivalence of categories
     Some technical notions
	Universal objects
	Direct and free products (products and coproducts)
	Other examples of universal objects
	Tensor products of modules
	Adjoint functors
	Cones, equalizers, and limits

THE CATEGORY OF SETS, TOPOI. FUZZY SETS
   Further general concepts.
	Remarks on the category of sets
	Amalgams and coamalgams. Cartesian squares.
	Completeness and cocompleteness
	Exponentiation
	Cartesian closed categories
   Topoi
	Subobjects
	Elements. Names of arrows.
	Subobject classifier.
	Definition of a topos
	Power objects
	General remarks on topoi. Well-pointed topoi.
	Examples of topoi
	Operations with subobjects. Heyting algebras.
	The Heyting algebra of subobjects. Boolean topoi.
   Fuzzy sets. Miscellany.
	Fuzzy sets and fuzzy quotient sets.
	The category of fuzzy sets
	The topos of fuzzy sets
	Remarks on the foundations. History.

VARIETIES OF ALGEBRAS. AXIOMATIZABLE CLASSES.
   Varieties.
	Closed classes and free algebras
	Classes and identities
	Birkhoff theorem
	Verbal functions
   Some constructions
	Free products in varieties
	Amalgams
	Epimorphisms and monomorphisms in varieties
   Axiomatic classes of algebras
  	General remarks
	Reduced products
	Quasivarieties and pseudovarieties

CATEGORY ALGEBRA AND ALGEBRAIC THEORIES
   Clones and clones of operations
	Clones of operations
	Abstract clones
	Clones and free algebras
	Representations of clones and varieties
   Algebraic theories
	Clones and categories
	Algebras as functors
	Algebraic theories and varieties of algebras
	Additional remarks

ALGEBRAIC LOGIC

BOOLEAN ALGEBRAS AND PROPOSITIONAL CALCULUS
   Boolean Algebras
	Boolean algebras, rings and lattices
	Homomorphisms, ideals and lattices
	Free Boolean algebras
	Finite Boolean algebras
   Propositional calculus and Boolean algebras
	Propositional calculus
	The Lindenbaum-Tarski algebra of propositional calculus
	Consistence, compatibility and models

HALMOS ALGEBRAS AND PREDICATE CALCULUS
   Halmos algebras
	Quantifiers and quantifier logic
	Halmos algebras
	Supports of elements
   Halmos algebras of predicate calculus
	Predicate calculus
	Halmos algebra of predicate calculus
   Halmos equality algebras. Cylindric algebras.
	Equality in Halmos algebras
	Cylindric algebras
   Homorphisms and structure of Halmos algebras.
	Homorphisms, ideals and filters
	Simple algebras.Semisimplicity theorem.
	Constants, terms and predicates
	Other remarks

SPECIALIZED HALMOS ALGEBRAS
   Halmos algebras over a variety of universal algebras
	Axiomatics and examples
	General information
	Equality in specialized algebras
   Halmos algebra over a free algebra of a variety
	Supports of elements of a free algebra
	Specialized algebra of formulas
	Halmos algebra over the free algebra of a variety
   Modification ofthe scheme
	Changing the variety
	The kernel of a passage to subvariety
	Additional remarks			

CONNECTIONS WITH MODEL THEORY
   Existence of models
	Preliminaries
	The main theorem
	Additional remarks
   Miscellany
	Consistency, compatibility and completeness
	Certain applications of the model existence theorem
	Classes and filters. The knowledge base of a model.

THE CATEGORIAL APPROACH TO ALGEBRAIC LOGIC
   Relation algebras
	Notes on quantifiers
	Definition of relation algebras
	Another approach
   Relational algebras associated with Halmos algebras
	The principle construction
	Algebras of relations
	Additional remarks
   Generalizations
	Generalized relational algebras
	Intuitionistic logic and models in toposes
	
DATABASES - ALGEBRAIC ASPECTS

ALGEBRAIC MODEL OF A DATABASE
   Universal databases
   The model
   Dynamic databases
   Generalizations

EQUIVALENCE AND REORGANIZATION OF DATABASES
   More about homomorphisms
   Equivalence and reconstruction
   Functional dependencies of relations
   Changing states. Storage and cleaning.

SYMMETRIES OF RELATIONS AND GALOIS THEORY OF DATABASES
   The Galois theory of relational algebras.
   Proofs in the pure Halmos algebra case.
   The general case. Some consequences.
   Galois theory of databases.

CONSTRUCTIONS IN DATABASE THEORY
   Constructions not changing the scheme
   Constructions modifying the scheme
   Supplement

DISCUSSION AND CONCLUSION
   Outcome and problem
   Problems of implementations
   History and sources.

BIBLIOGRAPHY