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Research and Exposition in Mathematics
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Volume 21
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K. Denecke, O. Lueders (eds.)
General Algebra and Discrete Mathematics
272 pages, soft cover, ISBN 3-88538-221-0, EUR 38.00, 1995
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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.
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Contents
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Preface, 1--2
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- 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