Вклад в теорию многообразий полугрупп тема диссертации и автореферата по ВАК РФ 01.01.06, доктор наук Ли Эдмонд В.Х.

• Ли Эдмонд В.Х.
• доктор наук
• 2020

Оглавление диссертации доктор 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

265

Введение диссертации (часть автореферата) на тему «Вклад в теорию многообразий полугрупп»

Introduction

In order to guide research and organize knowledge, we group algebras into varieties. This way of classification has been so successful that it has no serious competitor.

—McKenzie, McNulty, and Taylor (1987)

Recall that a variety is a class of algebras that is closed under the formation of homomorphic images, subalgebras, and arbitrary direct products. By the celebrated theorem of Birkhoff (1935), varieties are precisely equationally defined classes of algebras. The theory of varieties is one of the most central topics of universal algebra; the monographs of Burris and Sankappanavar (1981), Gratzer (1979), and McKenzie et al. (1987) are some standard references on the subject.

0.1 Varieties of semigroups

A semigroup is a nonempty set endowed with an associative binary operation. The algebraic theory of semigroups dates back to Suschkewitsch (1928) and has grown into an independent branch of algebra after the work of Clifford (1933, 1941), Dubreil (1941), Rees (1940), and many others. Refer to Hollings (2009) for a historical overview of the early development of the theory. Since the 1960s, varieties of semigroups—which include periodic varieties of groups—have been intensely investigated. Varieties of semigroups endowed with additional operations have also received much attention; see, for example, the monographs on varieties of groups (Neumann, 1967), varieties of inverse semigroups (Petrich, 1984), and varieties of completely regular semigroups (Petrich and

Reilly, 1999). Over the years, several surveys have been, or are in the process of being published on various aspects of these varieties, such as equational properties and the finite basis problem (Araujo, Araujo, Cameron, Lee, and Raminhos, 2020; Gupta and Krasilnikov, 2003; Lee and Volkov, 2020; Shevrin and Volkov, 1985; Volkov, 2000, 2001), structural properties of semigroups in varieties (Shevrin and Martynov, 1985; Shevrin and Sukhanov, 1989), lattices of varieties (Aizenshtat and Boguta, 1979; Evans, 1971; Shevrin, Vernikov, and Volkov, 2009; Vernikov, 2015), and algorithmic problems (Kharlampovich and Sapir, 1995).

0.2 Objectives and organization

The present thesis is a collection of results from over 50 peer reviewed articles that I have published since completing the Ph.D. degree at Simon Fraser University (Lee, 2002); in particular, results from 20 of these articles are selected to form the main content, while relevant results from other articles are stated without proof. The primary focus of the thesis is on varieties of semigroups, varieties of involution semigroups, and varieties of monoids that satisfy various important properties described in Section 0.3 below, and on some unexpected differences between lattices of these three types of varieties.

An overview of the results of the thesis, their motivation, and a general survey of the relevant literature are presented in Chapter 1. After some notation and background information are given in Chapter 2, detailed justifications of the main results are established in Chapters 3-14; these twelve chapters, divided into three parts (Parts I, II, and III), are devoted to semigroups, involution semigroups, and monoids, respectively.

For the convenience of the reader, a list of symbols and an index are provided at the end of the thesis. To distinguish varieties of the three types of algebras, let VsemC, Vinv£, and VmonC denote the variety of semigroups, the variety of involution semigroups, and the variety of monoids generated by C, respectively. It is unambiguous to let 0 denote the trivial variety regardless of algebra type.

0.3 Important varietal properties

The reader experienced with topics from the surveys cited in Section 0.1, especially Shevrin and Volkov (1985) and Volkov (2000, 2001), may choose to skip the present section in the first reading and proceed directly to Chapter 1 for the main results of the thesis.

Finiteness conditions are indisputably very important in many branches of mathematics—the theory of varieties is not an exception. For varieties of algebras, there are several finiteness properties the investigation of which is highly relevant and popular: a variety is

• finitely based if its equational theory is finitely axiomatizable;

• finitely generated if it is generated by a finite algebra;

• small if it contains finitely many subvarieties.

As observed by M.V. Sapir (1991), these three properties are independent in the sense that a variety that satisfies any two properties need not satisfy the third. Since an algebra and the variety it generates satisfy the same identities, it is unambiguous to say that an algebra satisfies a certain equational property when the variety it generates also satisfies the same property. For instance, an algebra is finitely based if the variety it generates is finitely based.

0.3.1 Finite basis problem

The finite basis problem—the investigation of which algebras are finitely based—is one of the most prominent research problems in universal algebra. Apart from being very natural by itself, this problem also has several interesting and unexpected connections with other topics of theoretical and practical importance, for example, feasible algorithms for membership in certain classes of formal languages (see Almeida, 1994) and classical number-theoretic conjectures such as the twin prime conjecture, Goldbach's conjecture, and the odd perfect number conjecture (Perkins, 1989). One approach to the finite basis problem is to classify non-finitely based algebras, and an effective method is to locate algebras that satisfy a stronger property: an algebra in a locally finite variety is inherently non-finitely based if every locally finite variety containing it is non-finitely based

(Murskil, 1965; Perkins, 1980). In particular, a finite algebra is non-finitely based whenever the variety it generates contains some inherently non-finitely based algebra. A finite algebra that is not inherently non-finitely based is weakly finitely based.

0.3.2 Hereditary finite basis property

A finitely based variety that satisfies the stronger property of having only finitely based subvarieties is hereditarily finitely based while an algebra is hereditarily finitely based if it generates a hereditarily finitely based variety. For varieties of associative rings, the hereditary finite basis property is also known as the Specht property in honor of Specht (1950), who importantly asked if every variety of associative rings over a field of characteristic zero is finitely based. This question was affirmatively answered by Kemer (1988). Refer to Belov (2010) for a survey of results related to the Specht property for rings and some other general algebras.

Since there can only be countably many finite sets of identities up to renaming of variables, a hereditarily finitely based variety can contain at most countably many subvarieties. Hereditarily finitely based varieties are closely related to minimal non-finitely based varieties, which are also called limit varieties. Indeed, since Zorn's lemma implies that each non-finitely based variety must contain some limit subvariety, a variety is hereditarily finitely based if and only if it excludes limit subvarieties.

0.3.3 Cross varieties

A variety is Cross if it is finitely based, finitely generated, and small. Being Cross is a hereditary property since every subvariety of each Cross variety is Cross (see Oates-MacDonald and Vaughan-Lee, 1978). Consequently, every Cross variety is hereditarily finitely based. A minimal non-Cross variety is commonly called an almost Cross variety. As in the case of non-finitely based varieties possessing limit subvarieties, it follows from Zorn's lemma that each non-Cross variety contains some almost Cross subvariety. Therefore a variety is Cross if and only if it excludes almost Cross subvarieties.

Заключение диссертации по теме «Математическая логика, алгебра и теория чисел», Ли Эдмонд В.Х.

Conclusion

The thesis is a systematic study of varieties of semigroups, varieties of involution semigroups, and varieties of monoids that satisfy various important properties, and of the differences between lattices of these three types of varieties.

Main achievements

(1) All aperiodic Rees-Suschkewitsch varieties are proved to be finitely based and those that are Cross, finitely generated, or small are completely characterized.

(2) Alternating identities that are hereditarily finitely based are completely described.

(3) General sufficient conditions are established under which a semigroup is non-finitely based or has no irredundant identity bases, and a new infinite class of finite semigroups without irredundant identity bases is exhibited.

(4) The interrelations between equational properties of involution semigroups and their semigroup reducts are examined in depth.

(a) It is first shown that a finite involution semigroup and its semigroup reduct can satisfy very different equational properties. In particular, three finite involution semigroups sharing the same semigroup reduct are constructed with the property that one has a finite identity basis, one has an infinite irredundant identity basis, and one has no irredundant identity bases.

(b) On the other hand, it is proved that for any involution semigroup whose variety contains some semilattice with nontrivial involution, the non-finite basis property is inherited from its semigroup reduct. This gives a powerful method in converting many results on equational properties of semigroups to results applicable to involution semigroups.

(5) The existence and uniqueness of two maximal hereditarily finitely based overcommutative varieties of monoids are established. This leads to a description of varieties of monoids from several large classes that are hereditarily finitely based. Varieties of aperiodic monoids with central idempotents that are Cross or inherently non-finitely generated are then completely characterized.

(6) A recipe is presented to construct the first examples of non-finitely based finite semigroups S with the property that the monoids S1 are finitely based.

Influence on other work

Results related to (1) and the techniques used to establish them have been employed in the investigation of several topics, for example, Rees-Suschkewitsch varieties with nontrivial groups (Kublanovskii, 2011; Lee and Volkov, 2011), varieties generated by aperiodic Rees-Suschkewitsch monoids (Lee, 2008b, 2009c, 2011b, 2012c), varieties generated by semigroups of order up to six (Edmunds et al., 2010; Lee, 2013b; Lee and Li, 2011; Lee and Zhang, 2015), and join irreducible pseudovarieties of finite semigroups (Lee, 2021; Lee et al., 2019, 2021).

The method mentioned in (4b) has been used in the solution of the finite basis problem for a number of involution semigroups (Ashikmin et al., 2015; Zhang et al., 2020) and in the construction of varieties of involution semigroups and varieties of involution monoids with several extreme properties (Lee, 2018b, 2019c).

Results in (5) are fundamental to both the construction of the first examples of finitely universal varieties of monoids (Gusev and Lee, 2020) and descriptions of the following varieties: limit varieties and almost Cross varieties of aperiodic monoids with commuting idempotents (Gusev, 2020b, 2021) and chain varieties of monoids (Gusev and Vernikov, 2018).

Applicability of methods used in thesis

Since the majority of results in the thesis is concerned with equational properties, naturally, syntactic methods are often employed. How applicable are these methods to the study of other types of finite algebras? Classical algebras such as groups (Oates and Powell, 1964), associative rings (Kruse, 1973; L'vov, 1973), and Lie rings (Bahturin and Ol'shanskil, 1975) are "too tamed" for these methods because finite members always generate Cross varieties. However, syntactic methods do work quite well for other types of algebras beyond those considered in the present thesis; see, for example, Dapic et al. (2007), Isaev and Kislitsin (2015), Jezek (2007), Lee (2009b), Lyndon (1954), and Murskii (1965).

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