Вклад в теорию многообразий полугрупп Ли Эдмонд В.Х.

Короткий опис

Ли Эдмонд В. Х.
Вклад в теорию многообразий полугрупп = Contributions to the theory of varieties of semigroups : Contributions to the theory of varieties of semigroups : диссертация ... доктора физико-математических наук : 01.01.06 / Edmond W. H. Lee; [Место защиты: ФГАОУ ВО «Национальный исследовательский университет «Высшая школа экономики»]. - Москва, 2020. - 300 с.
Оглавление диссертациидоктор наук Ли Эдмонд В.Х.
Contents
Page
Introduction
0.1 Varieties of semigroups
0.2 Objectives and organization
0.3 Important varietal properties
0.3.1 Finite basis problem
0.3.2 Hereditary finite basis property
0.3.3 Cross varieties
1 Historical overview and main results
1.1 Varieties generated by completely 0-simple semigroups
1.1.1 Rees-Suschkewitsch varieties
1.1.2 The varieties A2 and B2
1.1.3 Aperiodic Rees-Suschkewitsch varieties
1.1.4 Rees-Suschkewitsch varieties containing nontrivial groups
1.1.5 Aperiodic Rees-Suschkewitsch monoids
1.2 Hereditary finite basis property
1.2.1 Hereditarily finitely based identities
1.2.2 Alternating hereditarily finitely based identities
1.2.3 Minimal non-finitely based semigroups
1.3 Non-finite basis property
1.3.1 Finite basis problem for finite semigroups
1.3.2 Establishing the non-finite basis property
1.3.2.1 Critical Rees matrix semigroups
1.3.2.2 Inherently non-finitely based finite semigroups
1.3.2.3 Syntactic method
1.3.2.4 A comparison of the three methods
1.3.3 Irredundant identity bases
1.4 Varieties of involution semigroups
1.4.1 Equational properties of involution semigroups
1.4.2 Lattice of varieties of involution semigroups
1.4.3 Relationship between an involution semigroup and its semigroup reduct
1.4.3.1 Non-twisted involution semigroups
1.4.3.2 Inherent non-finite basis property
1.4.3.3 Sufficient conditions for the non-finite basis property
1.5 Varieties of monoids
1.5.1 Rees quotients of free monoids
1.5.2 Limit varieties and hereditarily finitely based varieties
1.5.3 Cross varieties and inherently non-finitely generated varieties
1.5.4 Further examples involving Rees quotients of free monoids
2 Preliminaries
2.1 Identities and deducibility
2.2 Identity bases and varieties
2.3 Connected words and identities
2.4 Rees quotients of free monoids
2.5 Involution semigroups
2.5.1 Terms, words, and plain words
2.5.2 Term identities and deducibility
I SEMIGROUPS
3 Aperiodic Rees-Suschkewitsch varieties
3.1 Background information on L(A2)
3.1.1 Identity bases of some subvarieties of A2
3.1.2 Identities defining varieties in [Ao, A2]
3.1.3 A decomposition of L(A2)
3.2 Finite basis property for subvarieties of A2
3.2.1 The interval 12 = [Ao,B2]
3.2.2 The interval Ii = [Ao V B2, A2]
3.2.3 The intervals I3 = B, Ao] and I4 = [Bo, Ao n B2]
3.2.4 The interval I5 = L(Bo)
3.3 The lattice L(A2)
3.3.1 The interval I5 = L(Bo)
3.3.2 The varieties D£, E, F, (£ ]t, and [ r )t
3.3.2.1 The varieties D
3.3.2.2 The varieties E and F
3.3.2.3 The varieties (£ ]i and [ r )t
3.3.3 The intervals Ii = [Ao V B2, A2] and 12 = [Ao, B2]
3.3.4 The intervals 13 = [B2, Ao] and 14 = [Bo, Ao n B2]
3.3.5 The interval [Bo, A2]
3.4 Subvarieties of A2 that are Cross, finitely generated, or small
3.4.1 Cross subvarieties and small subvarieties of A2
3.4.2 Finitely generated subvarieties of A2
4 A problem of Pollak and Volkov on hereditarily finitely based identities
4.1 Non-homotypical identities
4.2 Homotypical identities
5 Sufficient conditions for the non-finite basis property
5.1 Identities satisfied by
5.2 Proof of Theorem
5.3 Specialized versions of Theorem
6 Semigroups without irredundant identity bases
6.1 Sufficient condition for the absence of irredundant identity bases
6.2 Identities satisfied by L3,n
6.3 Sandwich identities
6.4 Restrictions on sandwich identities
6.4.1 Level of sandwiches forming sandwich identities
6.4.2 Refined sandwich identities
6.5 An explicit identity basis for L3,n
6.6 Absence of irredundant identity bases for L3,n
6.7 Variety membership problem for L3,n
Part I summary
II INVOLUTION SEMIGROUPS
7 Involution semigroups with infinite irredundant identity bases
7.1 Identities satisfied by (L3,n, )
7.2 Connected identities and *-sandwich identities
7.2.1 Connected identities
7.2.2 *-Sandwich identities
7.3 Restrictions on *-sandwich identities
7.3.1 Type of *-sandwiches forming *-sandwich identities
7.3.2 Refined *-sandwich identities
7.4 An explicit identity basis for (L3,n, ) with ^ >
7.4.1 An identity basis from (7.1)
7.4.2 A simpler identity basis
7.5 An infinite irredundant identity basis for (L3,n, ш ) with ^ >
7.5.1 The identities (7.4j)
7.5.2 Proof of Theorem
7.6 Smaller examples
8 Finitely based involution semigroups with non-finitely based reducts
8.1 Identities and *-sandwich identities satisfied by (L3,n, *1)
8.2 Restrictions on *-sandwich identities
8.3 An explicit identity basis for (L3,n, *1)
8.4 A finite identity basis for (L3,n, *1)
9 Counterintuitive examples of involution semigroups
9.1 Involution semigroups with different types of identity bases
9.1.1 Involution semigroups with irredundant identity bases
9.1.2 Involution semigroups without irredundant identity bases
9.2 Two incomparable chains of varieties of involution semigroups
9.2.1 The involution semigroups (L2n, *)
9.2.2 The involution semigroups (£2n+i; *)
9.2.3 Proof of Theorem
10 Equational theories of twisted involution semigroups
10.1 Organized identity bases
10.2 Proof of Theorem
Part II summary
III MONOIDS
11 Hereditarily finitely based varieties of monoids
11.1 Identities satisfied by noncommutative subvarieties of O
11.1.1 Canonical form
11.1.2 Fundamental identities and well-balanced identities
11.1.3 Proof of Proposition
11.2 Finite basis property of subvarieties of O
11.3 Distinguished varieties
12 Varieties of aperiodic monoids with central idempotents
12.1 Rigid identities
12.1.1 Definition and basic properties
12.1.2 Straubing identities
12.1.3 Limiting identities
12.2 The variety K
12.2.1 Almost Cross property
12.2.2 Subvarieties of K
12.3 Cross subvarieties of Azen
12.4 Varieties inherently non-finitely generated within Azen
12.5 A non-finitely generated subvariety of Rc{x2y2}
13 Certain Cross varieties of aperiodic monoids with commuting idempotents
13.1 The variety Q1
13.2 Varieties that contain Q1
13.3 Varieties that exclude K
13.4 Proof of Theorem
14 Counterintuitive examples of monoids
14.1 The direct product of Rc{xyx} with noncommutative groups of finite exponent
14.1.1 Identities satisfied by Rgfxyx} and by noncommutative groups
14.1.2 Proof of Theorem
14.2 Finitely based monoids from non-finitely based semigroups
Part III summary
Conclusion
Bibliography List of symbols Index
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