Research and Exposition in Mathematics

Volume 21

K. Denecke, O. Lueders (eds.)
General Algebra and Discrete Mathematics
272 pages, soft cover, ISBN 3-88538-221-0, EUR 38.00, 1995

Short Description

This volume contains articles based on lectures given at the "Fourth Conference on Discrete Mathematics", which took place at Potsdam in 1993.
The articles put in evidence some aspects of the natural interaction between General Algebra and Discrete Mathematics.
Algebraic structures such as semigroups, lattices, Boolean algebras, function algebras, and relation algebras, or ordered algebraic structures, form a structural background of such fields of Discrete Mathematics as formal languages, the theory of automata, theoretical computer science, and graph theory.
The distinction between discrete and non-discrete mathematics has perhaps something to do with the distinction between analog computers and digital computers. At any rate, the beginning of Discrete Mathematics as an own branch of mathematics is connected with the development of digital computers. Roughly, this distinction is analogous to the distinction between measuring and counting.
But all analog computers made by man have one serious defect; they do not measure accurately enough. The difficulty comes from the fact that the device records the continuous changes continuously. As a result there is always a very small ambiguity in its readings. A digital computer has no such defect. It is a machine to calculate numbers, not measuring phenomena. An analog signal has continuously valid interpretations. A digital signal has only a discrete number of valid interpretations, often a finite number. The digital signal is therefore always clear, never ambiguous; as a result calculations can be arranged to deliver exactly correct results. A finitary operation defined on a finite set models a digital device with a finite number of inputs and one output where a signal has only interpretations in this finite set. This model is one of the basic ingrediences of the papers presented in this volume.

Contents

Preface, 1--2

R. Bodendiek, G. Walter

On Number Theoretical Methods in Graph Labellings
3--26

A. Bulatov, A. Krokhin, K. Safin, E. Sukhanov

On the Structure of Clone Lattices,
27--34

I. Chajda

Congruence Properties of Algebras in Nilpotent Shifts of Varieties,
35--46

S. Dahlke

The Construction of Wavelets on Groups and Manifolds,
47--58

K. Denecke, D. Lau, R. Poeschel, D. Schweigert

Free Clones and Solid Varieties,
59--82

K. Denecke, J. Plonka

Regularization and Normalization of Solid Varieties,
83--92

D. Dimovski

On (m+k, m) - Groups for k < m,
93--100

J. Duda

d-fold Projections of Subalgebras, Homomorphisms, and Congruence Classes,
101--106

K. Gajewska-Kurdziel

On the Lattice of some Varieties Defined by Externally Compatible Identities,
107--110

E. Graczynska

Regular Identities H,
111-130

K. Halkowska

Free Algebras over P-Compatible Varieties,
131--136

H.-J. Hoehnke

On Certain Classes of Categories and Monoids Constructed from Abstract Mal'cev Clones, IV,
137--168

I. Korec

Decidable and Undecidable Theories of Generalized Pascal Triangles,
169--180

V. Levignon, S. E. Schmidt

A Geometric Approach to Generalized Matroid Lattices,
181--186

O. M. Mamedov

On the Lattice of Interpretability Types of Varieties,
187--190

I. Mirchev

Separable and Dominating Sets of Variables for Functions,
191--198

J. Plonka

On Hyperidentities of some Varieties,
199--214

M. Reichel

Free Spectra and Hyperidentities,
215--226

H.-J. Vogel

On Quasivarieties Generated by Diagonal-Inversion-Algebras,
227--242

W. Wessel

Are All Complete Plane Multimaps But One Bounded by Euler Only?,
243--272