Combinatorial Group Theory: COMBINATORIAL GROUP THEORY

The Algebraic Theory of Semigroups, Vol. I. By A. H. Clifford and G. B. Preston. Pp. xv + 224. $10.60. 1961. (American Mathematical Society, Providence.)

Discoveries in finite semigroups have influenced several mathematical fields, including theoretical computer science, tropical algebra via matrix theory with coefficients in semirings, and other areas of modern algebra. This comprehensive, encyclopedic text will provide the reader - from the graduate student to the researcher/practitioner with a detailed understanding of modern finite semigroup theory, focusing in particular on advanced topics on the cutting edge of research. Key features: (1) Develops q-theory, a new theory that provides a unifying approach to finite semigroup theory via quantization; (2) Contains the only contemporary exposition of the complete theory of the complexity of finite semigroups; (3) Introduces spectral theory into finite semigroup theory; (4) Develops the theory of profinite semigroups from first principles, making connections with spectra of Boolean algebras of regular languages; (5) Presents over 70 research problems, most new, and hundreds of exercises. Additional features: (1) For newcomers, an appendix on elementary finite semigroup theory; (2) Extensive bibliography and index. The q-theory of Finite Semigroups presents important techniques and results, many for the first time in book form, and thereby updates and modernizes the literature of semigroup theory.

The q-theory of Finite Semigroups

Roughly speaking, a variety of groups is a class of groups in which certain relations called laws or rules or identities are universally valid. The most widely investigated variety of groups is that of abelian groups in which the commutative law is universally valid or, in other words, in which for arbitrary elements x1, x2 of the group, the identity 

$$ \left( {{x_1},{x_2}} \right) = 1 $$

holds, where, as usual, (x1, x2) is defined as the commutator 

$$ x_1^{ - 1}x_2^{ - 1}{x_1}{x_2} $$

of x1 and x2. Similarly, the Burnside variety of exponent e is defined as the class of groups in which, for all elements x, the relation 

$$ {x^e} = 1 $$

holds. The metabelian groups of Chapter II.6 can be defined as the variety of groups define4 by the identity 

$$ \left( {\left( {{x_1},{x_2}} \right),\left( {{x_3},{x_4}} \right)} \right) = 1, $$

and the nilpotent groups of class c discussed in Chapter II.7 form the variety defined by the identity 

$$ \left( {{x_1},{x_2},...,{x_c},{x_{c + 1}}} \right) = 1 $$

where the parentheses ( ) denote the simple (c + l)-fold commutator defined in Chapter II.7.

Varieties of Groups

This book is about relations between three different areas of mathematics and theoretical computer science: combinatorial group theory, cryptography, and complexity theory. It explores how non-commutative (infinite) groups, which are typically studied in combinatorial group theory, can be used in public-key cryptography. It also shows that there is remarkable feedback from cryptography to combinatorial group theory because some of the problems motivated by cryptography appear to be new to group theory, and they open many interesting research avenues within group theory. In particular, a lot of emphasis in the book is put on studying search problems, as compared to decision problems traditionally studied in combinatorial group theory. Then, complexity theory, notably generic-case complexity of algorithms, is employed for cryptanalysis of various cryptographic protocols based on infinite groups, and the ideas and machinery from the theory of generic-case complexity are used to study asymptotically dominant properties of some infinite groups that have been applied in public-key cryptography so far. This book also describes new interesting developments in the algorithmic theory of solvable groups and another spectacular new development related to complexity of group-theoretic problems, which is based on the ideas of compressed words and straight-line programs coming from computer science.

Non-Commutative Cryptography and Complexity of Group-Theoretic Problems

Equational theories of semigroups with involution

A family of pseudovarieties of solvable groups is constructed, each of which has decidable membership and undecidable extension problem for partial permutations. Included are a pseudovariety U satisfying no non-trivial group identity and a metabelian pseudovariety Q. For each of these pseudovarieties V, the inverse monoid pseudovariety Sl*V has undecidable membership problem. As a consequence, it is proved that the pseudovariety operators *, **, ?, =, = n , and P do not preserve decidability. In addition, several joins, including A V U, are shown to be undecidable.

https://www.ams.org/proc/2003-131-09/S0002-9939-03-06860-6/S0002-9939-03-06860-6.pdf

On the extension problem for partial permutations

Abstract. It is proved that the series of all Brauer monoids generates the pseudovariety of all finite monoids while the series of their aperiodic analogues, the Jones monoids (also called Temperly–Lieb monoids), generates the pseudovariety of all finite aperiodic monoids. The proof is based on the analysis of wreath product decomposition and Krohn–Rhodes theory. The fact that the Jones monoids form a generating series for the pseudovariety of all finite aperiodic monoids can be viewed as solution of an old problem popularized by J.-É. Pin. For the latter, the relationship between the Jones monoids and the monoids of order preserving mappings of a chain of length n is investigated.

Pseudovarieties generated by Brauer type monoids

We study matrix identities involving multiplication and unary operations such as transposition or Moore-Penrose inversion. We prove that in many cases such identities admit no finite basis.

https://www.ems-ph.org/journals/show_pdf.php?issn=1435-9855&vol=14&iss=3&rank=11

Matrix identities involving multiplication and transposition

We prove a sufficient condition under which a semigroup admits no finite identity basis. As an application, it is shown that the identities of the Kauffman monoid $${\mathcal{K}_n}$$
 are nonfinitely based for each $${n \geq 3}$$
 . This result holds also for the case when $${\mathcal{K}_n}$$
 is considered as an involution semigroup under either of its natural involutions.

Karl Auinger

Papers

Equational theories of semigroups with involution

On the extension problem for partial permutations

Pseudovarieties generated by Brauer type monoids

Matrix identities involving multiplication and transposition

The finite basis problem for Kauffman monoids