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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
Scientific council: DH 01-01.01.06-27.03.08
Institute of Mathematics and Computer Science

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

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