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On linear and inverse quasigroups and their applications in code theory


Author: Şcerbacov Victor
Degree:doctor habilitat of physics and mathematics
Speciality: 01.01.06 - Mathematical logics, algebra and theory of numbers
Year:2008
Institution: Institute of Mathematics and Computer Science of the ASM
Scientific council: DH 01-01.01.06-27.03.08
Institute of Mathematics and Computer Science of the ASM

Status

The thesis was presented on the 17 April, 2008
Approved by NCAA on the 19 June, 2008

Abstract

Adobe PDF document – 0.30 Mb / in romanian

Thesis

CZU 512.548

Adobe PDF document 1.16 Mb / in english
247 pages

Keywords

quasigroup, n-ary quasigroup, linear quasigroup, inverse quasigroup, medial quasigroup, Latin square, code, automorphism, orthogonality



Summary

This thesis is devoted to the theory of n-ary and binary quasigroups and their applications in code theory.

The following results are obtained:

The thesis is written in Englis

Summary


1 Introduction
  • 1.1 Introduction and main results
    • 1.1.1 Outline of the history and development of the topics considered in this dissertation
    • 1.1.2 Summary of the contents of the dissertation
  • 1.2 Main definitions, notions and concepts
    • 1.2.1 Binary groupoids, quasigroups and loops
    • 1.2.2 Squares and Latin squares
    • 1.2.3 Isotopy of groupoids and parastrophy of quasigroups
    • 1.2.4 Translations of groupoids and quasigroups, isostrophisms
    • 1.2.5 Some definitions and elementary properties of quasigroups
    • 1.2.6 Generators of inner multiplication groups

2 Binary inverse quasigroups
  • 2.1 Inverse quasigroups and generalized identities
    • 2.1.1 Definitions of various inverse quasigroups
    • 2.1.2 Autostrophisms .
    • 2.1.3 Generalized balanced parastrophic identities
  • 2.2 Construction and properties of (r, s, t)-inverse quasigroups
    • 2.2.1 Elementary properties and examples .
    • 2.2.2 Left-linear quasigroups which are (r, s, t)-inverse
    • 2.2.3 Main theorems
    • 2.2.4 Direct products of (r,s,t)-quasigroups
    • 2.2.5 The existence of (r,s,t)-inverse quasigroups
    • 2.2.6 Weak-inverse-property quasigroups
  • 2.3 Nuclei of inverse quasigroups
    • 2.3.1 A-nuclei of quasigroups
    • 2.3.2 Nuclei of λ- and ρ-inverse quasigroups
    • 2.3.3 Nuclei of (α, β, γ)-inverse quasigroups
3 On Burmistrovich-Belousov and Bruck-Belousov problems
  • 3.1 On quasigroups with Moufang identity
  • 3.2 Bruck-Belousov problem
    • 3.2.1 Introduction .
    • 3.2.2 Congruences of a quasigroup and its associated group
    • 3.2.3 Normality of congruences of some inverse quasigroups
    • 3.2.4 On behavior of congruences by an isotopy
4 On the structure of n-ary medial quasigroups
  • 4.1 On n-ary medial quasigroups
    • 4.1.1 n-ary quasigroups, their isotopy and translations
    • 4.1.2 Linear n-ary quasigroups
    • 4.1.3 n-ary medial quasigroups
    • 4.1.4 Homomorphisms of n-ary quasigroups
    • 4.1.5 Direct product of n-ary quasigroups .
    • 4.1.6 Homomorphisms of n-ary linear quasigroups
    • 4.1.7 On n-ary analog of Murdoch Theorem
  • 4.2 On structure of n-ary simple medial quasigroups
    • 4.2.1 Simple n-ary quasigroups
    • 4.2.2 Congruences of linear n-ary quasigroups
    • 4.2.3 On n-ary simple medial quasigroups
  • 4.3 Solvability of finite n-ary medial quasigroups
  • 4.4 On binary medial quasigroups
    • 4.4.1 On Toyoda Theorem
    • 4.4.2 Examples
    • 4.4.3 On (m,n)-elements
    • 4.4.4 On (m,n)-linear quasigroups

5 Autotopies and automorphisms of quasigroups
  • 5.1 On autotopies and automorphisms of n-ary linear quasigroups
    • 5.1.1 Autotopies and automorphisms of derivative groups
    • 5.1.2 Automorphisms of n-ary T-quasigroups
    • 5.1.3 Automorphisms of some isotopes of quasigroups
    • 5.1.4 Automorphism groups of n-ary medial quasigroups
    • 5.1.5 Examples
  • 5.2 Automorphism groups of some binary quasigroups
    • 5.2.1 Automorphisms of some loop isotopes
    • 5.2.2 Automorphism groups of left distributive quasigroups
    • 5.2.3 Automorphism groups of isotopes of left distributive quasigroups

6 Quasigroups and codes
  • 6.1 Codes with one check symbol from a quasigroup point of view
    • 6.1.1 Introduction
    • 6.1.2 On possibilities of quasigroup codes
    • 6.1.3 Totally anti-commutative quasigroups and n-quasigroup codes
    • 6.1.4 5-n-quasigroup codes
    • 6.1.5 Phonetic errors
    • 6.1.6 Examples of codes
    • 6.1.7 About the system of the serial numbers of German banknotes
  • 6.2 On signs of Bol loop translations

7 Orthogonality of binary quasigroups
  • 7.1 Orthogonality. Introduction
    • 7.1.1 m-Tuples of maps and its product
    • 7.1.2 Groupoids and m-tuples of maps, kinds of tuples
    • 7.1.3 The τ-property of m-tuples of permutations
    • 7.1.4 Definitions of orthogonality
    • 7.1.5 Orthogonality in works of V.D. Belousov
    • 7.1.6 Product of squares
  • 7.2 Orthogonality and parastrophe orthogonality
    • 7.2.1 Orthogonality of left quasigroups
    • 7.2.2 Orthogonality of quasigroups and its parastrophes
    • 7.2.3 Orthogonality in the language of quasi-identities
    • 7.3 Transformations which preserve orthogonality
    • 7.3.1 Isotopy and (12)-isostrophy
    • 7.3.2 On generalized isotopy of squares
    • 7.3.3 Gisotopy and orthogonality
  • 7.4 Orthogonality of T-quasigroups
    • 7.4.1 Parastrophic orthogonality
    • 7.4.2 (12)-parastrophe orthogonality