We survey several results and open problems related to synchronizing automata. In particular, we discuss some recent advances towards a solution of the Cerný conjecture.

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Synchronizing Automata and the Černý Conjecture

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

Set theory is the most fundamental part of mathematics. The definition of almost any kind of mathematical object (a group, a ring, a vector space, a topological space, a Hilbert space...) begins: a consists of a set, together with some extra structure in the form of operations, relations, subsets, sets of subsets, functions to the real numbers, or whatever. Also, as we will see, these operations, relations, etc. are themselves special kinds of sets.

Naïve set theory

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Classical finite transformation semigroups

https://u.cs.biu.ac.il/~margolis/Representation%20Theory%20Seminar/Monoid%20Representations%20and%20Applications/Solomon%20Rook%20presentation.pdf

Representations of the rook monoid

Fundamentals Free inverse semigroups and the theorems of McAlister Biordered sets Zigzags and their applications Semigroup diagrams and word problems Combinatorial aspects of transformation semigroups References Glossary of notation Index.

Techniques of semigroup theory

For a semigroup S and a set the relative rank of S modulo A is the minimal cardinality of a setB such that generates S. We show that the relative rank of an infinite full transformation semigroup modulo the symmetric group, and also modulo the set of all idempotent mappings, is equal to 2. We also characterise all pairs of mappings which, together with the symmetric group or the set of all idempotents, generate the full transformation semigroup.

On relative ranks of full transformation semigroups

The monoid of orientation-preserving mappings on a chain

Consider the finite set Xn = {1,2, …,n} ordered in the standard way. Let Tn denote the full transformation semigroup on Xn, that is, the semigroup of all mappings α: Xn→Xn under composition. We shall call α order-preserving if i ≤ j implies iα ≤ jα for i,j∈Xn, and α is decreasing if iα ≤ i for all i∈Xn. This paper investigates combinatorial properties of the semigroup O of all order-preserving mappings on Xn, and of its subsemigroup C which consists of all decreasing and order-preserving mappings.

Combinatorial results for semigroups of order-preserving mappings

For a semigroup S and a set A ⊆ S the relative rank of S modulo A is the minimal cardinality of a set B such that A∪B generates S; we denote this by rank(S : A). We also allow ourselves to call this the relative rank of A in S. In [8] the authors showed that if we take S = TX , an infinite full transformation semigroup, then the relative rank of S modulo A is 2 in the cases where A is either the full symmetric group or the set of all idempotents of S. On the other hand if A is itself a countable set and A ∪B generates TX , then B must be uncountable as TX is itself uncountable. This prompts the question as to whether it is possible to find a subset A of TX which is of countable relative rank greater than 2. Surprisingly, the answer is ‘no’. That conclusion can be deduced from Proposition 1.1 that was first proved by W. Sierpinski in 1935 [10]. A simpler proof was immediately furnished however by Stefan Banach [1]. Nonetheless, this result seems not widely known as is evidenced by the fact that the often quoted result of Evans [2], published in 1952, that any countable semigroup can be embedded in a two-generator semigroup follows at once from Sierpinski’s result. For that reason, and in order to make the paper more selfcontained, we reproduce the short proof of Banach.

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Peter M. Higgins

Papers

Techniques of semigroup theory

On relative ranks of full transformation semigroups

The monoid of orientation-preserving mappings on a chain

Combinatorial results for semigroups of order-preserving mappings

Countable versus uncountable ranks in infinite semigroups of transformations and relations