Finite Semigroup Theory Research Articles

An algebraic characterization of self-generating chemical reaction networks using semigroup models

The ability of a chemical reaction network to generate itself by catalyzed reactions from constantly present environmental food sources is considered a fundamental property in origin-of-life research. Based on Kaufmann’s autocatalytic sets, Hordijk and Steel have constructed the versatile formalism of catalytic reaction systems (CRS) to model and to analyze such self-generating networks, which they named reflexively autocatalytic and food-generated. Recently, it was established that the subsequent and simultaenous catalytic functions of the chemicals of a CRS give rise to an algebraic structure, termed a semigroup model. The semigroup model allows to naturally consider the function of any subset of chemicals on the whole CRS. This gives rise to a generative dynamics by iteratively applying the function of a subset to the externally supplied food set. The fixed point of this dynamics yields the maximal self-generating set of chemicals. Moreover, the set of all functionally closed self-generating sets of chemicals is discussed and a structure theorem for this set is proven. It is also shown that a CRS which contains self-generating sets of chemicals cannot have a nilpotent semigroup model and thus a useful link to the combinatorial theory of finite semigroups is established. The main technical tool introduced and utilized in this work is the representation of the semigroup elements as decorated rooted trees, allowing to translate the generation of chemicals from a given set of resources into the semigroup language.

The sympathetic sceptic’s guide to semigroup representations

This is an elementary introduction to the representation theory of finite semigroups. We illustrate the Clifford–Munn correspondence between the representations of a semigroup and the representations of its maximal subgroups. The emphasis throughout is on naturally occurring examples.

On the Dowling and Rhodes lattices and wreath products

Dowling and Rhodes defined different lattices on the set of triples (Subset, Partition, Cross Section) over a fixed finite group G. Although the Rhodes lattice is not a geometric lattice, it defines a matroid in the sense of the theory of boolean representable simplicial complexes. This turns out to be the direct sum of a uniform matroid with the lift matroid of the complete gain link graph over G. As is well known, the Dowling lattice defines the frame matroid over a similar gain graph. This gives a new perspective on both matroids and also an application of matroid theory to the theory of finite semigroups. We also make progress on an important question for these classical matroids: what are the minimal boolean representations and the minimum degree of a boolean matrix representation?

SOME IMPORTANT APPLICATIONS OF SEMIGROUPS

This Paper deals with the some important applications of semigroups in general and regular semigroups in particular.The theory of finite semigroups has been of particular importance in theoretical computer science since the 1950s because of the natural link between finite semigroups and finite automata via the syntactic monoid. In probability theory, semigroups are associated with Markov process. In section 2 we have seen different areas of applications of semigroups. We identified some Applications in biology, Partial Differential equation, Formal Languages etc whose semigroup structures are nothing but regular.

Join irreducible semigroups

We begin a systematic study of finite semigroups that generate join irreducible members of the lattice of pseudovarieties of finite semigroups, which are important for the spectral theory of this lattice. Finite semigroups [Formula: see text] that generate join irreducible pseudovarieties are characterized as follows: whenever [Formula: see text] divides a direct product [Formula: see text] of finite semigroups, then [Formula: see text] divides either [Formula: see text] or [Formula: see text] for some [Formula: see text]. We present a new operator [Formula: see text] that preserves the property of join irreducibility, as does the dual operator, and show that iteration of these operators on any nontrivial join irreducible pseudovariety leads to an infinite hierarchy of join irreducible pseudovarieties. We also describe all join irreducible pseudovarieties generated by a semigroup of order up to five. It turns out that there are [Formula: see text] such pseudovarieties, and there is a relatively easy way to remember them. In addition, we survey most results known about join irreducible pseudovarieties to date and generalize a number of results in Sec. 7.3 of [The[Formula: see text]-theory of Finite Semigroups, Springer Monographs in Mathematics (Springer, Berlin, 2009)].

Pseudovarieties of Ordered Completely Regular Semigroups

This paper is a contribution to the theory of finite semigroups and their classification in pseudovarieties, which is motivated by its connections with computer science. The question addressed is what role can play the consideration of an order compatible with the semigroup operation. In the case of unions of groups, so-called completely regular semigroups, the problem of which new pseudovarieties appear in the ordered context is solved. As applications, it is shown that the lattice of pseudovarieties of ordered completely regular semigroups is modular and that taking the intersection with the pseudovariety of bands defines a complete endomorphism of the lattice of all pseudovarieties of ordered semigroups.

Unified theory for finite Markov chains

We provide a unified framework to compute the stationary distribution of any finite irreducible Markov chain or equivalently of any irreducible random walk on a finite semigroup S. Our methods use geometric finite semigroup theory via the Karnofsky–Rhodes and the McCammond expansions of finite semigroups with specified generators; this does not involve any linear algebra. The original Tsetlin library is obtained by applying the expansions to P(n), the set of all subsets of an n element set. Our set-up generalizes previous groundbreaking work involving left-regular bands (or R-trivial bands) by Brown and Diaconis, extensions to R-trivial semigroups by Ayyer, Steinberg, Thiéry and the second author, and important recent work by Chung and Graham. The Karnofsky–Rhodes expansion of the right Cayley graph of S in terms of generators yields again a right Cayley graph. The McCammond expansion provides normal forms for elements in the expanded S. Using our previous results with Silva based on work by Berstel, Perrin, Reutenauer, we construct (infinite) semaphore codes on which we can define Markov chains. These semaphore codes can be lumped using geometric semigroup theory. Using normal forms and associated Kleene expressions, they yield formulas for the stationary distribution of the finite Markov chain of the expanded S and the original S. Analyzing the normal forms also provides an estimate on the mixing time.

Merge Decompositions, Two-sided Krohn–Rhodes, and Aperiodic Pointlikes

AbstractThis paper provides short proofs of two fundamental theorems of finite semigroup theory whose previous proofs were significantly longer, namely the two-sided Krohn-Rhodes decomposition theorem and Henckell’s aperiodic pointlike theorem. We use a new algebraic technique that we call the merge decomposition. A prototypical application of this technique decomposes a semigroup $T$ into a two-sided semidirect product whose components are built from two subsemigroups $T_{1}$, $T_{2}$, which together generate $T$, and the subsemigroup generated by their setwise product $T_{1}T_{2}$. In this sense we decompose $T$ by merging the subsemigroups $T_{1}$ and $T_{2}$. More generally, our technique merges semigroup homomorphisms from free semigroups.

Semigroups of rectangular matrices under a sandwich operation

Let $\mathcal M_{mn}=\mathcal M_{mn}(\mathbb F)$ denote the set of all $m\times n$ matrices over a field $\mathbb F$, and fix some $n\times m$ matrix $A\in\mathcal M_{nm}$. An associative operation $\star$ may be defined on $\mathcal M_{mn}$ by $X\star Y=XAY$ for all $X,Y\in\mathcal M_{mn}$, and the resulting \emph{sandwich semigroup} is denoted $\mathcal M_{mn}^A=\mathcal M_{mn}^A(\mathbb F)$. These semigroups are closely related to Munn rings, which are fundamental tools in the representation theory of finite semigroups. In this article, we study $\mathcal M_{mn}^A$ as well as its subsemigroups $\operatorname{Reg}(\mathcal M_{mn}^A)$ and $\mathcal E_{mn}^A$ (consisting of all regular elements and products of idempotents, respectively), as well as the ideals of $\operatorname{Reg}(\mathcal M_{mn}^A)$. Among other results, we: characterise the regular elements, determine Green's relations and preorders, calculate the minimal number of matrices (or idempotent matrices, if applicable) required to generate each semigroup we consider, and classify the isomorphisms between finite sandwich semigroups $\mathcal M_{mn}^A(\mathbb F_1)$ and $\mathcal M_{kl}^B(\mathbb F_2)$. Along the way, we develop a general theory of sandwich semigroups in a suitably defined class of \emph{partial semigroups} related to Ehresmann-style "arrows only" categories, we hope this framework will be useful in studies of sandwich semigroups in other categories. We note that all our results have applications to the \emph{variants} $\mathcal M_n^A$ of the full linear monoid $\mathcal M_n$ (in the case $m=n$), and to certain semigroups of linear transformations of restricted range or kernel (in the case that $\operatorname{rank}(A)$ is equal to one of $m,n$).

On the atoms of algebraic lattices arising in 𝔮-theory

We determine many of the atoms of the algebraic lattices arising in [Formula: see text]-theory of finite semigroups.

Book review of “The $$q$$ q -theory of finite semigroups”

Book review of “The $$q$$ q -theory of finite semigroups”

John Rhodes and the theory of finite semigroups

John Rhodes and the theory of finite semigroups

The Join of the Varieties of R-trivial and L-trivial Monoids via Combinatorics on Words

Automata, Logic and Semantics The join of two varieties is the smallest variety containing both. In finite semigroup theory, the varieties of R-trivial and L-trivial monoids are two of the most prominent classes of finite monoids. Their join is known to be decidable due to a result of Almeida and Azevedo. In this paper, we give a new proof for Almeida and Azevedo's effective characterization of the join of R-trivial and L-trivial monoids. This characterization is a single identity of omega-terms using three variables.

On free spectra of finite monoids from the pseudovariety DA

The pseudovariety DA consists of all aperiodic finite monoids all of whose regular Open image in new window-classes are subsemigroups (that is, rectangular subbands); this pseudovariety appears quite frequently in various contexts in finite semigroup theory. In this note we prove that all its members have a log-polynomial free spectrum, thereby making a new step towards proving the Seif conjecture on the dichotomy of free spectra of finite monoids.

Representation theory of finite semigroups over semirings

We develop the representation theory of a finite semigroup over an arbitrary commutative semiring with unit, in particular classifying the irreducible and minimal representations. The results for an arbitrary semiring are as good as the results for a field. Special attention is paid to the boolean semiring, where we also characterize the simple representations and introduce the beginnings of a character theory.

An upper bound for the power pseudovariety PCS

It is a celebrated result in finite semigroup theory that the equality of pseudovarieties PG = BG holds, where PG is the pseudovariety of finite monoids generated by all power monoids of finite groups and BG is the pseudovariety of all block groups, that is, the pseudovariety of all finite monoids all of whose regular \({\fancyscript{D}}\)-classes have the property that the corresponding principal factors are inverse semigroups. Moreover, it is well known that BG = JⓜG, where JⓜG is the pseudovariety of finite monoids generated by the Mal’cev product of the pseudovarieties J and G of all finite \({\fancyscript{J}}\)-trivial monoids and of all finite groups, respectively. In this paper, a more general kind of finite semigroups is considered; namely, the so-called aggregates of block groups are introduced. It follows that the class AgBG of all aggregates of block groups forms a pseudovariety of finite semigroups. It is next proved that AgBG = JⓜCS, where JⓜCS is the pseudovariety of finite semigroups generated by the Mal’cev product of the pseudovarieties J and CS, whilst, this once, J stands for the pseudovariety of all finite \({\fancyscript{J}}\)-trivial semigroups and CS stands for the pseudovariety of all finite completely simple semigroups. Furthermore, it is shown that the power pseudovariety PCS, which is the pseudovariety of finite semigroups generated by all power semigroups of finite completely simple semigroups, has the property that \({{\bf PCS}\subseteq{\bf AgBG}}\). However, the question whether this inclusion is strict or not is left open.

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 Černý’s conjecture for an important class of automata.

Definability of Languages by Generalized First-Order Formulas over $(\mathbb{N},+)$

We consider an extension of first-order logic by modular quantifiers of a fixed modulus $q$. Drawing on collapse results from finite model theory and techniques of finite semigroup theory, we show that if the only available numerical predicate is addition, then sentences in this logic cannot define the set of bit strings in which the number of 1's is divisible by a prime $p$ that does not divide $q$. More generally, we completely characterize the regular languages definable in this logic. The corresponding statement, with addition replaced by arbitrary numerical predicates, is equivalent to the conjectured separation of the circuit complexity class $ACC$ from $NC^1$. Thus our theorem can be viewed as proving a highly uniform version of the conjecture.

Learning in varieties of the form [formula omitted] from positive data

Grammatical inference and finite semigroup theory are continuously developing. Nevertheless it seems that they are not interacting too much. We propose in this paper an inference method for languages that belong to varieties of the form V * LI . Many well known families of languages like locally testable, reversible, dot-depth one, etc. are of that form. The method unifies existing algorithms for inference of some of those families and can be applied to some others that had not been inferred yet. It uses a result about the cascade product in which one of the factors is a transducer and the second is the automaton obtained by inferring the base case of the family using the transduction of the sample as input.

Möbius functions and semigroup representation theory

This paper explores several applications of Möbius functions to the representation theory of finite semigroups. We extend Solomon's approach to the semigroup algebra of a finite semilattice via Möbius functions to arbitrary finite inverse semigroups. This allows us to explicitly calculate the orthogonal central idempotents decomposing an inverse semigroup algebra into a direct product of matrix algebras over group rings. We also extend work of Bidigare, Hanlon, Rockmore and Brown on calculating eigenvalues of random walks associated to certain classes of finite semigroups; again Möbius functions play an important role.