M Solid Varieties of Algebras Advances in Mathematics 1st Edition by Koppitz, Klaus Denecke ISBN 0387308040 978-0387308043

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Product details: 

ISBN 10:  0387308040

ISBN 13:  978-0387308043

Author:  J. Koppitz,  Klaus Denecke

M-Solid Varieties of Algebras provides a complete and systematic introduction to the fundamentals of the hyperequational theory of universal algebra, offering the newest results on M-solid varieties of semirings and semigroups. The book aims to develop the theory of M-solid varieties as a system of mathematical discourse that is applicable in several concrete situations. It applies the general theory to two classes of algebraic structures, semigroups and semirings. Both these varieties and their subvarieties play an important role in computer science.

A unique feature of this book is the use of Galois connections to integrate different topics. Galois connections form the abstract framework not only for classical and modern Galois theory, involving groups, fields and rings, but also for many other algebraic, topological, ordertheoretical, categorical and logical theories. This concept is used throughout the whole book, along with the related topics of closure operators, complete lattices, Galois closed subrelations and conjugate pairs of completely additive closure operators.

Table of contents: 

1 Basic Concepts

1.1 Subalgebras and Homomorphic Images

1.2 Direct and Subdirect Products

1.3 Term Algebras, Identities, Free Algebras

1.4 The Galois Connection (Id, Mod)

2 Closure Operators and Lattices

2.1 Closure Operators and Kernel Operators

2.2 Complete Sublattices of a Complete Lattice

2.3 Galois Connections and Complete Lattices

2.4 Galois Closed Subrelations

2.5 Conjugate Pairs of Additive Closure Operators

3 M-Hyperidentities and M-solid Varieties

3.1 M-Hyperidentities

3.2 The Closure Operators XM, XM

3.3 M-Solid Varieties and their Characterization

3.4 Subvariety Lattices and Monoids of Hypersubstitutions

3.5 Derivation of M-Hyperidentities

4 Hyperidentities and Clone Identities

4.1 Menger Algebras of Rank n

4.2 The Clone of a Variety

5 Solid Varieties of Arbitrary Type

5.1 Rectangular Algebras

5.2 Solid Chains

6 Monoids of Hypersubstitutions

6.1 Basic Definitions

6.2 Injective and Bijective Hypersubstitutions

6.3 Finite Monoids of Hypersubstitutions of Type (2)

6.4 The Monoid of all Hypersubstitutions of Type (2)

6.5 Green’s Relations on Hyp(2).

6.6 Idempotents in Hyp(2, 2)

6.7 The Order of Hypersubstitutions of Type (2, 2)

6.8 Green’s Relations in Hyp(n, n)

6.9 The Monoid of Hypersubstitutions of Type (n)

6.10 Left-Seminearrings of Hypersubstitutions

7 M-Solid Varieties of Semigroups

7.1 Basic Concepts on M-Solid Varieties of Semigroups

7.2 Regular-solid Varieties of Semigroups

7.3 Solid Varieties of Semigroups

7.4 Pre-solid Varieties of Semigroups

7.5 Locally Finite and Finitely Based M-solid Varieties

8 M-solid Varieties of Semirings

8.1 Necessary Conditions for Solid Varieties of Semirings

8.2 The Minimal Solid Variety of Semirings.

8.3 The Greatest Solid Variety of Semirings

8.4 The Lattice of all Solid Varieties of Semirings

8.5 Generalization of Normalizations

8.6 All Pre-solid Varieties of Semirings

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