Representation Theory of Finite Semigroups, Semigroup Radicals and Formal Language Theory

In this paper we characterize the congruence associated to the direct sum of all irreducible representations of a finite semigroup over an arbitrary field, generalizing results of Rhodes for the field of complex numbers. Applications are given to obtain many new results, as well as easier proofs of several results in the literature, involving: triangularizability of finite semigroups; which semigroups have (split) basic semigroup algebras, two-sided semidirect product decompositions of finite monoids; unambiguous products of rational languages; products of rational languages with counter; and \v{C}ern\'y's conjecture for an important class of automata

Effective dimension of finite semigroups

In this paper we discuss various aspects of the problem of determining the minimal dimension of an injective linear representation of a finite semigroup over a field. We outline some general techniques and results, and apply them to numerous examples.Comment: To appear in J. Pure Appl. Al

Quivers of monoids with basic algebras

We compute the quiver of any monoid that has a basic algebra over an algebraically closed field of characteristic zero. More generally, we reduce the computation of the quiver over a splitting field of a class of monoids that we term rectangular monoids (in the semigroup theory literature the class is known as $\mathbf{DO}$) to representation theoretic computations for group algebras of maximal subgroups. Hence in good characteristic for the maximal subgroups, this gives an essentially complete computation. Since groups are examples of rectangular monoids, we cannot hope to do better than this. For the subclass of $\mathscr R$-trivial monoids, we also provide a semigroup theoretic description of the projective indecomposables and compute the Cartan matrix.Comment: Minor corrections and improvements to exposition were made. Some theorem statements were simplified. Also we made a language change. Several of our results are more naturally expressed using the language of Karoubi envelopes and irreducible morphisms. There are no substantial changes in actual result

Representation theory of finite semigroups, semigroup radicals and formal language theory

In this paper we characterize the congruence associated to the direct sum of all irreducible representations of a finite semigroup over an arbitrary field, generalizing results of Rhodes for the field of complex numbers. Applications are given to obtain many new results, as well as easier proofs of several results in the literature, involving: triangularizability of finite semigroups; which semigroups have (split) basic semigroup algebras, two-sided semidirect product decompositions of finite monoids; unambiguous products of rational languages; products of rational languages with counter; andČerný's conjecture for an important class of automata

Some fundamental properties for regular element of electroencephalography signals semigroup during epileptic seizure

Electroencephalography (EEG) is a record of electrical activity along the scalp. It is measures voltage fluctuations resulting from ionic current flows within the neurons of the brain. EEG is most often used to diagnose epilepsy, which causes understandable abnormalities in EEG readings. The mathematical analysis of EEG signals assists medical specialists by providing a clarification of the brain activity being observed, so increasing the information about the structure and function of the human brain. EEG signals during epileptic seizure can be viewed as a semigroup of square matrices under matrix multiplication, and every element in that semigroup is shown to be regular. In this paper, we will present some main properties of regular element of EEG signals during epileptic seizure

M\"obius Functions and Semigroup Representation Theory II: Character formulas and multiplicities

We generalize the character formulas for multiplicities of irreducible constituents from group theory to semigroup theory using Rota's theory of M\"obius inversion. The technique works for a large class of semigroups including: inverse semigroups, semigroups with commuting idempotents, idempotent semigroups and semigroups with basic algebras. Using these tools we are able to give a complete description of the spectra of random walks on finite semigroups admitting a faithful representation by upper triangular matrices over the complex numbers. These include the random walks on chambers of hyperplane arrangements studied by Bidigare, Hanlon, Rockmere, Brown and Diaconis. Applications are also given to decomposing tensor powers and exterior products of rook matrix representations of inverse semigroups, generalizing and simplifying earlier results of Solomon for the rook monoid.Comment: Some minor typos corrected and references update

Recent advances in algorithmic problems for semigroups

In this article we survey recent progress in the algorithmic theory of matrix semigroups. The main objective in this area of study is to construct algorithms that decide various properties of finitely generated subsemigroups of an infinite group $G$, often represented as a matrix group. Such problems might not be decidable in general. In fact, they gave rise to some of the earliest undecidability results in algorithmic theory. However, the situation changes when the group $G$ satisfies additional constraints. In this survey, we give an overview of the decidability and the complexity of several algorithmic problems in the cases where $G$ is a low-dimensional matrix group, or a group with additional structures such as commutativity, nilpotency and solvability.Comment: survey article for SIGLOG New