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Status
The thesis was presented on the 17 April, 2008 Approved by NCAA on the 19 June, 2008
Abstract
– 0.30 Mb / in romanian
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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:
- New classes of binary inverse ( (r,s,t)-inverse, (α,β,γ)-inverse) quasigroups are introduced, their properties are researched.
- It is proved that quasigroup with any Moufang identity is a loop.
- Some progress in the solving of Bruck--Belousov problem on normality of congruences of quasigroups is achieved for left (right) loops.
- Description of structure of $n$-ary simple medial quasigroups is given.
- The structure of finite $n$-ary medial quasigroups is given.
- Automorphism groups of n-ary T-quasigroups, n-ary medial quasigroups and some isotopes of binary left distributive quasigroups are researched.
- New families of easy constructed codes with one check symbol, which have characteristics better than known codes of such kind, are discovered.
- Necessary and sufficient conditions of orthogonality of a finite quasigroup and any its parastrophe are given.
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