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Mathematics - Algebra | M-Solid Varieties of Algebras

M-Solid Varieties of Algebras

Series: Advances in Mathematics, Vol. 10

Koppitz, Jörg, Denecke, Klaus

2006, XIII, 341 p.

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  • About this book

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.

Audience

This book is intended for researchers in the fields of universal algebra, semigroups, and semirings; researchers in theoretical computer science; and students and lecturers in these fields.

Content Level » Research

Keywords » Algebraic structure - Clone identity - Hyperidentity - Hypersubstitution - M-solid variety - Menger Algebra - Monoid

Related subjects » Algebra - Software Engineering - Theoretical Computer Science

Table of contents / Sample pages 

Preface Chapter 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) Chapter 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 Chapter 3 M-Hyperidentities and M-solid Varieties 3.1 M-Hyperidentities 3.2 The Closure Operators 3.3 M-Solid Varieties and their Characterization 3.4 Subvariety Lattices and Monoids of Hypersubstitutions 3.5 Derivation of M-Hyperidentities Chapter 4 Hyperidentities and Clone Identities 4.1 Menger Algebras of Rank n 4.2 The Clone of a Variety Chapter 5 Solid Varieties of Arbitrary Type 5.1 Rectangular Algebras 5.2 Solid Chains Chapter 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 Chapter 7 M-Solid Varieties of Semigroups 7.1 Basic Concepts onM-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 Chapter 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 Bibliography Glossary Index

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