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# Вклад в теорию многообразий полугруппCONTRIBUTIONS TO THE THEORY OF VARIETIES OF SEMIGROUPS

Соискатель:
Ли Эдмонд В.Х.
Члены комитета:
Беклемишев Лев Дмитриевич (Математический институт им. В.А. Стеклова РАН, д.ф.-м.н, председатель комитета), Stuart W. Margolis (Стюарт Марголис) (Bar-Ilan University, Israel, PhD, член комитета), Белов Алексей Яковлевич (Bar-Ilan University, Israel, д.ф.-м.н, член комитета), Верников Борис Муневич (Уральский федеральный университет имени первого Президента России Б. Н. Ельцина, д.ф.-м.н, член комитета), Красильников Алексей Николаевич (Университет города Бразилиа, Бразилия, д.ф.-м.н, член комитета)
Диссертация принята к предварительному рассмотрению:
4/10/2020
Диссертация принята к защите:
5/21/2020
Дисс. совет:
Совет по математике
Дата защиты:
9/30/2020
The primary focus of the thesis is on varieties of semigroups and of related algebras such as involution semigroups and monoids. Main results on varieties of semigroups include a positive solution to the finite basis problem for all aperiodic Rees–Suschkewitsch varieties and a complete description of all such varieties that are either Cross or finitely generated. Sufficient conditions are also established under which a semigroup has no finite identity bases or even irredundant ones.
For varieties of involution semigroups, the emphasis is on investigating how involution semigroups are related to their semigroup reducts with respect to equational properties. An important positive result is concerned with involution semigroups whose varieties contain some semilattice with nontrivial unary operation: such an involution semigroup is non-finitely based whenever its semigroup reduct is non-finitely based. For a negative result, a trio of finite involution semigroups sharing a common semigroup reduct is constructed so that one has a finite identity basis, one has an infinite irredundant identity basis, and one has no irredundant identity bases.
Regarding varieties of monoids, the majority of results are concerned with varieties of aperiodic monoids with central idempotents; most notably, characterizations of those that are hereditarily finitely based, Cross, or inherently non-finitely generated are established. The first examples of non-finitely based finite semigroups that generate finitely based monoids are also constructed.
Диссертация [*.pdf, 2.56 Мб] (дата размещения 7/27/2020)
Резюме [*.pdf, 326.56 Кб] (дата размещения 7/27/2020)
Summary [*.pdf, 584.78 Кб] (дата размещения 7/27/2020)

#### Публикации, в которых излагаются основные результаты диссертации

E.W.H. Lee. Locally finite monoids in finitely based varieties
E.W.H. Lee. A sufficient condition for the absence of irredundant bases.
M. Jackson, E.W.H. Lee. Monoid varieties with extreme properties
E.W.H. Lee. Equational theories of unstable involution semigroups
E.W.H. Lee. Finite involution semigroups with infinite irredundant bases of identities
E.W.H. Lee. On certain Cross varieties of aperiodic monoids with commuting idempotents
E.W.H. Lee. Maximal Specht varieties of monoids.
E.W.H. Lee. A sufficient condition for the non-finite basis property of semigroups.
E.W.H. Lee. Combinatorial Rees–Sushkevich varieties that are Cross, finitely generated, or small
E.W.H. Lee. Combinatorial Rees–Sushkevich varieties are finitely based

Сведения о результатах защиты:
Комитет по диссертации рекомендовал присудить ученую степень доктора наук (Протокол № 2 от 30.09.2020 г.). Решением диссертационного совета НИУ ВШЭ по математике (Протокол № 5 от 30.10.2020 г.) присуждена ученая степень доктора математических наук.
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