The Theory of Partial Algebraic Operations

1. Front Matter

   Pages i-x

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2. Basic Terminology

   • E. S. Ljapin, A. E. Evseev

   Pages 1-3

3. Initial Concepts and Properties

   • E. S. Ljapin, A. E. Evseev

   Pages 5-27

4. Homomorphisms

   • E. S. Ljapin, A. E. Evseev

   Pages 29-67

5. Divisibility Relations

   • E. S. Ljapin, A. E. Evseev

   Pages 69-88

6. Intermediate Associativity

   • E. S. Ljapin, A. E. Evseev

   Pages 89-102

7. Semigroup Extensions of Partial Operations

   • E. S. Ljapin, A. E. Evseev

   Pages 103-142

8. Partial Groupoids of Transformations

   • E. S. Ljapin, A. E. Evseev

   Pages 143-174

9. Factorisation of Partial Groupoids

   • E. S. Ljapin, A. E. Evseev

   Pages 175-213

10. Back Matter

    Pages 215-238

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Nowadays algebra is understood basically as the general theory of algebraic oper­ ations and relations. It is characterised by a considerable intrinsic naturalness of its initial notions and problems, the unity of its methods, and a breadth that far exceeds that of its basic concepts. It is more often that its power begins to be displayed when one moves outside its own limits. This characteristic ability is seen when one investigates not only complete operations, but partial operations. To a considerable extent these are related to algebraic operators and algebraic operations. The tendency to ever greater generality is amongst the reasons that playa role in explaining this development. But other important reasons play an even greater role. Within this same theory of total operations (that is, operations defined everywhere), there persistently arises in its different sections a necessity of examining the emergent feature of various partial operations. It is particularly important that this has been found in those parts of algebra it brings together and other areas of mathematics it interacts with as well as where algebra finds applica­ tion at the very limits of mathematics. In this connection we mention the theory of the composition of mappings, category theory, the theory of formal languages and the related theory of mathematical linguistics, coding theory, information theory, and algebraic automata theory. In all these areas (as well as in others) from time to time there arises the need to consider one or another partial operation.

• algebra
• associative property
• functional analysis
• group theory
• information
• information theory
• proof

• Russian State Pedagogical University, St Petersburg, Russia

  E. S. Ljapin, A. E. Evseev

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