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Lattices of Formations of Algebraic Structures

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Author(s)
Tsarev, Aleksandr
Issued Date
2020
URI
http://dcoll.jejunu.ac.kr/common/orgView/000000009489
Abstract
본 논문에서는 대수 구조의 형성에 대한 다양한 격자를 조사 연구하였다. 유한군들의 클래스가 형성(formation)이라는 정의는 클래스의 한 원소인 군 G에 대하여 그의 상군 G/N을 포함하고, 만약 G/N1과 G/N2가 클래스에 속한다면 G/N1 ∩ N2도 포함하는 조건을 만족하는 클래스이다. 본 논문에서의 주요결과는 다음과 같다. 다양한 국소 형성들에 대응하는 언어들을 설명하였다. σ는 모든 소수의 집합의 분할이라 하자. 모든 형성에 대한 격자의 모든 법칙은 모든 다양한 σ-국소형성들에 대한 격자 안에서 만족된다는 것을 증명하였다. 모든 함자 닫힌 합성 형성의 격자는 대수 격자라는 것과 모든 함자 닫힌 형성에 대한 격자의 모든 법체계가 모든 함자 닫힌 곱셈 부분 합성 형성들의 격자와 법체계가 일치한다는 것을 보였다. 또한, 모든 X-국소형성의 격자는 대수적이고 모듈러 격자라 는 것을 증명하였다. M을 T-부분군의 최소성과 최대성 조건들을 만족하는 모든 복합연산자 T-군의 클래스라 하자. 그러면, 모든 함자 닫힌 M-형성에 대한 격자의 모든 법칙은 모든 함자 닫힌 곱셈 부분 엽층 형성들에 대한 격자 안에서 만족된다는 것을 증명하였다.
In the dissertation, various lattices of formations of algebraic structures are investi-gated. The languages corresponding to multiply local formations are described. Let σ be a partition of the set of all primes. It is proved that every law of the lattice of all formations is fullled in the lattice of all multiply σ-local formations. It is shown that the lattice of all functor-closed totally composition formations is algebraic, and that the law system of the lattice of all functor-closed formations coincides with the law system of the lattice of all functor-closed multiply partially composition forma-tions. It is proved that the lattice of all X-local formations is algebraic and modular. Let M be the class of all multioperator T-groups satisfying the minimality and max-imality conditions for T-subgroups. It is proved that every law of the lattice of all functor-closed M-formations is fullled in the lattice of all functor-closed multiply partially foliated M-formations. Keywords: Monoid, Language, Group, Ring, Multioperator T-Group, Fuzzy Set, Lattice, Formation, Saturated Formation, Local Formation, Composition Formation Mathematics Subject Classication (2020) 20F17, 20D10, 20M35, 68Q70, 03E72 UDC 512.542
Affiliation
제주대학교 대학원
Department
대학원 수학과
Advisor
송석준
Awarded Date
2020. 8
Table Of Contents
1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 The initial idea . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 Research contribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2. Formations of Algebraic Structures . . . . . . . . . . . . . . . . . . . . 10
2.1 Literature review & background . . . . . . . . . . . . . . . . . . . . . . . 10
2.2 Formations of monoids and formal languages . . . . . . . . . . . . . . . 12
2.2.1 Classes of monoids . . . . . . . . . . . . . . . . . . . . . . . 12
2.2.2 Formations of languages . . . . . . . . . . . . . . . . . . . . . 18
2.3 Formations of nite groups . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.3.1 Semigroup of all formations . . . . . . . . . . . . . . . . . . 20
2.3.2 τ -Closed formations . . . . . . . . . . . . . . . . . . . . . . . 22
2.3.3 Lattices of formations . . . . . . . . . . . . . . . . . . . . . . 24
2.3.4 Saturated formations . . . . . . . . . . . . . . . . . . . . . . . 25
2.3.5 Multiply local formations . . . . . . . . . . . . . . . . . . . . 27
2.3.6 σ-Local formations . . . . . . . . . . . . . . . . . . . . . . . 29
2.3.7 Solvably saturated formations . . . . . . . . . . . . . . . . . . 31
2.3.8 Multiply ω-composition formations . . . . . . . . . . . . . . 33
2.3.9 X-local formations . . . . . . . . . . . . . . . . . . . . . . . . 33
2.4 Formations of rings and some generalizations . . . . . . . . . . . . . . . 35
2.4.1 Classes of nite rings . . . . . . . . . . . . . . . . . . . . . . 37
2.4.2 Lattices of formations of nite rings . . . . . . . . . . . . . . 39
2.4.3 Classes of T-groups . . . . . . . . . . . . . . . . . . . . . . . 44
2.4.4 1-Foliated formations . . . . . . . . . . . . . . . . . . . . . 47
3. Lattices of Saturated Formations and Group Languages . . . . . . . . 51
3.1 Languages associated with local formations . . . . . . . . . . . . . . . . . 51
3.2 Lattices of σ-local formations . . . . . . . . . . . . . . . . . . . . . . . . 55
3.2.1 Laws of the lattices of multiply σ-local formations . . . . . . 56
3.2.2 Frattini subformations . . . . . . . . . . . . . . . . . . . . . . 60
3.2.3 Languages associated with σ-local formations . . . . . . . . . 62
4. Lattices of Partially Composition Formations of Finite Groups . . . . 64
4.1 Inductive lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
4.1.1 Minimal satellites . . . . . . . . . . . . . . . . . . . . . . . . 65
4.1.2 Inductance of the lattice c . . . . . . . . . . . . . . . . . . 71
4.1.3 Inductance of the lattice c. . . . . . . . . . . . . . . . . . 74
2.3.6 σ-Local formations . . . . . . . . . . . . . . . . . . . . . . . 29
2.3.7 Solvably saturated formations . . . . . . . . . . . . . . . . . . 31
2.3.8 Multiply ω-composition formations . . . . . . . . . . . . . . 33
2.3.9 X-local formations . . . . . . . . . . . . . . . . . . . . . . . . 33
2.4 Formations of rings and some generalizations . . . . . . . . . . . . . . . 35
2.4.1 Classes of nite rings . . . . . . . . . . . . . . . . . . . . . . 37
2.4.2 Lattices of formations of nite rings . . . . . . . . . . . . . . 39
2.4.3 Classes of T-groups . . . . . . . . . . . . . . . . . . . . . . . 44
2.4.4 1-Foliated formations . . . . . . . . . . . . . . . . . . . . . 47
3. Lattices of Saturated Formations and Group Languages . . . . . . . . 51
3.1 Languages associated with local formations . . . . . . . . . . . . . . . . . 51
3.2 Lattices of σ-local formations . . . . . . . . . . . . . . . . . . . . . . . . 55
3.2.1 Laws of the lattices of multiply σ-local formations . . . . . . 56
3.2.2 Frattini subformations . . . . . . . . . . . . . . . . . . . . . . 60
3.2.3 Languages associated with σ-local formations . . . . . . . . . 62
4. Lattices of Partially Composition Formations of Finite Groups . . . . 64
4.1 Inductive lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
4.1.1 Minimal satellites . . . . . . . . . . . . . . . . . . . . . . . . 65
4.1.2 Inductance of the lattice c . . . . . . . . . . . . . . . . . . 71
4.1.3 Inductance of the lattice c . . . . . . . . . . . . . . . . . . 74
2.3.6 σ-Local formations . . . . . . . . . . . . . . . . . . . . . . . 29
2.3.7 Solvably saturated formations . . . . . . . . . . . . . . . . . . 31
2.3.8 Multiply ω-composition formations . . . . . . . . . . . . . . 33
2.3.9 X-local formations . . . . . . . . . . . . . . . . . . . . . . . . 33
2.4 Formations of rings and some generalizations . . . . . . . . . . . . . . . 35
2.4.1 Classes of nite rings . . . . . . . . . . . . . . . . . . . . . . 37
2.4.2 Lattices of formations of nite rings . . . . . . . . . . . . . . 39
2.4.3 Classes of T-groups . . . . . . . . . . . . . . . . . . . . . . . 44
2.4.4 1-Foliated formations . . . . . . . . . . . . . . . . . . . . . 47
3. Lattices of Saturated Formations and Group Languages . . . . . . . . 51
3.1 Languages associated with local formations . . . . . . . . . . . . . . . . . 51
3.2 Lattices of σ-local formations . . . . . . . . . . . . . . . . . . . . . . . . 55
3.2.1 Laws of the lattices of multiply σ-local formations . . . . . . 56
3.2.2 Frattini subformations . . . . . . . . . . . . . . . . . . . . . . 60
3.2.3 Languages associated with σ-local formations . . . . . . . . . 62
4. Lattices of Partially Composition Formations of Finite Groups . . . . 64
4.1 Inductive lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
4.1.1 Minimal satellites . . . . . . . . . . . . . . . . . . . . . . . . 65
4.1.2 Inductance of the lattice c . . . . . . . . . . . . . . . . . . 71
4.1.3 Inductance of the lattice c . . . . . . . . . . . . . . 74
4.2 Algebraic lattices of formations . . . . . . . . . . . . . . . . . . . . . . . 76
4.3 Separated lattices of formations . . . . . . . . . . . . . . . . . . . . . . . 80
4.4 Laws of the lattices of composition formations . . . . . . . . . . . . . . 87
4.5 Comments & further research . . . . . . . . . . . . . . . . . . . . . . . 90
5. Lattices of X-Local Formations of Finite Groups . . . . . . . . . . . . 93
5.1 Algebraic lattices of X-local formations . . . . . . . . . . . . . . . . . . . 93
5.2 Modular lattices of X-local formations . . . . . . . . . . . . . . . . . . . 100
6. Lattices of Formations of Multioperator T-groups . . . . . . . . . . . 103
6.1 Laws of the lattices of foliated formations . . . . . . . . . . . . . . . . . 103
6.2 Frattini subformations of foliated formations . . . . . . . . . . . . . . . . 108
7. Possible Future Directions . . . . . . . . . . . . . . . . . . . . . . . . . 114
7.1 Formations of formal languages . . . . . . . . . . . . . . . . . . . . . . . 115
7.2 Classes of fuzzy languages . . . . . . . . . . . . . . . . . . . . . . . . . . 117
7.3 Applications in programming and data science . . . . . . . . . . . . . . . 120
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
국문초록 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
Degree
Doctor
Publisher
제주대학교 대학원
Citation
Tsarev, Aleksandr. (2020). Lattices of Formations of Algebraic Structures
Type
Dissertation
Appears in Collections:
General Graduate School > Mathematics
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