Structure Of Semigroups Research Articles

Graphs and their associated inverse semigroups

Directed graphs have long been used to gain an understanding of the structure of semigroups, and recently the structure of directed graph semigroups has been investigated resulting in a characterization theorem and an analog of Frucht’s Theorem. We investigate two inverse semigroups defined over undirected graphs constructed from the notions of subgraph and vertex set induced subgraph. We characterize the structure of the semilattice of idempotents and lattice of ideals of these inverse semigroups. We prove a characterization theorem that states that every graph has a unique associated inverse semigroup up to isomorphism allowing for an algebraic restatement of the Edge Reconstruction Conjecture.

On the structure of quantum Markov semigroups of weak coupling limit type

We discuss some recent results on the structure of quantum Markov semigroups of weak coupling limit type and their stationary states. In particular, we identify the minimal central projection in the fixed point algebra where they act in a trivial way and show, when they admit a single Bohr frequency, that all invariant states are convex combinations of equilibrium states and arbitrary states supported in the above minimal central projection.

Cost and dimension of words of zero topological entropy

Let A * denote the free monoid generated by a finite nonempty set A. In this paper we introduce a new measure of complexity of languages L ⊆ A * defined in terms of the semigroup structure on A *. For each L ⊆ A * , we define its cost c(L) as the infimum of all real numbers α for which there exist a language S ⊆ A * with p S (n) = O(n α) and a positive integer k with L ⊆ S k. We also define the cost dimension d c (L) as the infimum of the set of all positive integers k such that L ⊆ S k for some language S with p S (n) = O(n c(L)). We are primarily interested in languages L given by the set of factors of an infinite word x = x 0 x 1 x 2 • • • ∈ A N of zero topological entropy, in which case c(L) 2 there exist infinite words x of positive cost and of complexity p x (n) = O(n α).

On the semigroups of order-preserving transformations generated by idempotents of rank n −1

Let [Formula: see text] be the semigroup of all singular order-preserving mappings on the finite set [Formula: see text]. It is known that [Formula: see text] is generated by its set of idempotents of rank [Formula: see text], and its rank and idempotent rank are [Formula: see text] and [Formula: see text], respectively. In this paper, we study the structure of the semigroup generated by any nonempty subset of idempotents of rank [Formula: see text] in [Formula: see text]. We also calculate its rank and idempotent rank.

On regularity and the word problem for free idempotent generated semigroups

The category of all idempotent generated semigroups with a prescribed structure $\mathcal{E}$ of their idempotents $E$ (called the biordered set) has an initial object called the free idempotent generated semigroup over $\mathcal{E}$, defined by a presentation over alphabet $E$, and denoted by $\mathsf{IG}(\mathcal{E})$. Recently, much effort has been put into investigating the structure of semigroups of the form $\mathsf{IG}(\mathcal{E})$, especially regarding their maximal subgroups. In this paper we take these investigations in a new direction by considering the word problem for $\mathsf{IG}(\mathcal{E})$. We prove two principal results, one positive and one negative. We show that, for a finite biordered set $\mathcal{E}$, it is decidable whether a given word $w \in E^*$ represents a regular element; if in addition one assumes that all maximal subgroups of $\mathsf{IG}(\mathcal{E})$ have decidable word problems, then the word problem in $\mathsf{IG}(\mathcal{E})$ restricted to regular words is decidable. On the other hand, we exhibit a biorder $\mathcal{E}$ arising from a finite idempotent semigroup $S$, such that the word problem for $\mathsf{IG}(\mathcal{E})$ is undecidable, even though all the maximal subgroups have decidable word problems. This is achieved by relating the word problem of $\mathsf{IG}(\mathcal{E})$ to the subgroup membership problem in finitely presented groups.

Dynamic Semi-Group CIA Pattern Optimizing the Risk on RTS

The preventive control is one of the best well advance control for recent complex IS Security Application to protect the data and services from the uncertainty, hacker, and unauthorized users. Now, increasing the demand and importance of business, information & communication system & growing the external risks is a very common phenomenon for everywhere. The RTS security put forward to the management focus on IT infrastructure. This work contributes to the development of an optimization pattern that aims to determine the optimal cost to be apply into security mechanisms deciding on the measure components of system security and resources. The author's mechanism should be design in such way, the Confidentiality, Integrity, Availability, Authenticity and Accountability are automatically PDC for all the time. The author has to optimize the system attacks and down time by implementing semi-group structure CIA pattern, mean while improving the throughput of the Business, Resources & Technology. Finally, the author has to maximize the protection of IT resources & Services for all the time and every time. This proposed CIA Pattern is the part of protection, detection, benchmarking, fault analysis and risk assessment of real time operating system and applicable to efficient resource management on web application.

A Structure of Some Abundant Semigroups

Semigroup theory is widely applied in many disciplines such as computer science, operation theory and combinatoric mathematics etc.In this paper, we introduce the concept of perfect ♯-abundant semigroups of type C and we overcome the difficulty about results on ♯-abundant which is a little than abundant semigroups,A construction theorem for such semigroups will be given.

Amenability and uniqueness for groupoids associated with inverse semigroups

We investigate recent uniqueness theorems for reduced \(C^*\)-algebras of Hausdorff etale groupoids in the context of inverse semigroups. In many cases the distinguished subalgebra is closely related to the structure of the inverse semigroup. In order to apply our results to full \(C^*\)-algebras, we also investigate amenability. More specifically, we obtain conditions that guarantee amenability of the universal groupoid for certain classes of inverse semigroups. These conditions also imply the existence of a conditional expectation onto a canonical subalgebra.

Semigroup structures and properties of some kinds of mappings

In this paper, we consider several kinds of maps frequently appearing in the population dynamics, e.g., logistic mapping: $f(x)=rx(1-x)$ $(x\in \mathbb{R}, $ $r>0$ is a parameter), unimodel Allee maps with Allee effect and Sigmoid mapping, etc. We study the persistence of some properties for these maps such as the numbers of fixed point, their positions, stabilities and so on. We prove that they are closed under composition of maps. That means they form a semigroup. The examples are given to illustrate the results.

On norms of operators generated by shift transformations arising in signal and image processing on meshes supplied with semigroup structures

Shift transformations and linear operators generated by shifts have a number of applications in signal and image processing. This note is concerned with a problem which has arisen in studying properties of real-world signals and images defined on meshes. For processing we suggest to introduce in domains of signals and images different semigroup structures. Semigroup operations give us opportunities to introduce shift transformations of signals and images. We study norms of polynomial filters generated by shift operators.

The structure of certain F-abundant semigroups

An abundant semigroup is called F-abundant if each class modulo the minimum cancellative congruence has a maximum element with respect to the natural order. This paper studies a kind of F-abundant semigroup in which all these maximum elements form a sub-semigroup and the set of idempotents is a semilattice. After the characterizations of such semigroup are analysed, the structure theorem is obtained by the semidirect product of a semilattice and a cancellative monoid.

Rees matrix covers for a class of locally U-commutative semigroups

ABSTRACTGiven the importance of Morita theory of semigroups, we continue the study on the local structure of semigroups. Here we consider a class of nonregular semigroups, called locally U-commutative semigroups having U-local units, containing the classes of locally inverse semigroups, locally adequate semigroups, locally Ehresmann semigroups, and semigroups with local units having locally commuting idempotents. Our aim is to give a Rees matrix covering theorem for such semigroups with a partial McAlister sandwich bundle, and hence to put all the existing results into one context.

Malcev product of superabundant semigroups and its subsemigroups

We use Malcev product of semigroups satisfying some axiomatic conditions to describe the structure of superabundant semigroups and some of its subclasses. Some characterization theorems of these kinds of semigroups are given.

On the biordered set of rings

In [4] K.S.S. Nambooripad introduced biordered sets as a partial algebra $\left(E, \omega^r, \omega^l\right)$ where $\omega^r$ and $\omega^l$ are two quasiorders on the set $E$ satisfying biorder axioms; to study the structure of a regular semigroup. Later in [2] David Esdown showed that the set of idempotents of a regular semigroup forms a regular biordered set. Here we extend the idea of biordered sets into rings and discussed some of its properties.

Weakly almost periodic functions, model-theoretic stability, and minimality of topological groups

We investigate the automorphism groups of $\aleph _0$-categorical structures and prove that they are exactly the Roelcke precompact Polish groups. We show that the theory of a structure is stable if and only if every Roelcke uniformly continuous function on the automorphism group is weakly almost periodic. Analysing the semigroup structure on the weakly almost periodic compactification, we show that continuous surjective homomorphisms from automorphism groups of stable $\aleph _0$-categorical structures to Hausdorff topological groups are open. We also produce some new WAP-trivial groups and calculate the WAP compactification in a number of examples.

Several Remarks on Groups of Automorphisms of Free Groups

Let $G$ be the group of automorphisms of a free group $F_\infty$ of infinite order. Let $H$ be the stabilizer of first $m$ generators of $F_\infty$. We show that the double cosets of $\Gamma$ with respect to $H$ admit a natural semigroup structure. For any compact group $K$ the semigroup $\Gamma$ acts in the space $L^2$ on the product of $m$ copies of $K$

THE SEMIGROUP OF METRIC MEASURE SPACES AND ITS INFINITELY DIVISIBLE PROBABILITY MEASURES.

A metric measure space is a complete, separable metric space equipped with a probability measure that has full support. Two such spaces are equivalent if they are isometric as metric spaces via an isometry that maps the probability measure on the first space to the probability measure on the second. The resulting set of equivalence classes can be metrized with the Gromov-Prohorov metric of Greven, Pfaffelhuber and Winter. We consider the natural binary operation ⊞ on this space that takes two metric measure spaces and forms their Cartesian product equipped with the sum of the two metrics and the product of the two probability measures. We show that the metric measure spaces equipped with this operation form a cancellative, commutative, Polish semigroup with a translation invariant metric. There is an explicit family of continuous semicharacters that is extremely useful for, inter alia, establishing that there are no infinitely divisible elements and that each element has a unique factorization into prime elements. We investigate the interaction between the semigroup structure and the natural action of the positive real numbers on this space that arises from scaling the metric. For example, we show that for any given positive real numbers a, b, c the trivial space is the only space that satisfies a ⊞ b = c . We establish that there is no analogue of the law of large numbers: if X1, X2, … is an identically distributed independent sequence of random spaces, then no subsequence of [Formula: see text] converges in distribution unless each Xk is almost surely equal to the trivial space. We characterize the infinitely divisible probability measures and the Lévy processes on this semigroup, characterize the stable probability measures and establish a counterpart of the LePage representation for the latter class.

The structure of a graph inverse semigroup

Given any directed graph E one can construct a graph inverse semigroup G(E), where, roughly speaking, elements correspond to paths in the graph. In this paper we study the semigroup-theoretic structure of G(E). Specifically, we describe the non-Rees congruences on G(E), show that the quotient of G(E) by any Rees congruence is another graph inverse semigroup, and classify the G(E) that have only Rees congruences. We also find the minimum possible degree of a faithful representation by partial transformations of any countable G(E), and we show that a homomorphism of directed graphs can be extended to a homomorphism (that preserves zero) of the corresponding graph inverse semigroups if and only if it is injective.

Structure of Endomorphism Semigroup of Endomappings of Some Particular Rooted Trees

For an endomapping of a finite set of points lying on some rooted particular trees, endomorphism and endomorphism semigroup were studied. The main aim of this paper was to obtain the structure of the semigroup of all endomorphisms of the endomappings represented by directed graphs on particular types of rooted trees.Journal of Bangladesh Academy of Sciences, Vol. 39, No. 2, 169-175, 2015

Malcev Products of Monoids and Varieties of Bands

As a generalization of completely regular semigroups, which can be written as \({\mathcal{(G \circ RB) \circ S}}\) where \({\mathcal{G}}\), \({\mathcal{RB}}\) and \({\mathcal S}\) are the varieties of groups, rectangular bands and semilattices, respectively, we have replaced \({\mathcal G}\) by the class \({\mathcal M}\) of monoids. This calls for finding the structure of such semigroups, and, as a first step, characterizations.