Simple Semigroups Research Articles

Congruence-Simple Acts Over Completely Simple Semigroups

Congruence-Simple Acts Over Completely Simple Semigroups

Principal right congruences on completely 0-simple semigroups

Principal right congruences on completely 0-simple semigroups

The number of spanning trees of E-extended power graphs associated with finite semigroups

We introduce a new kind of graph built from finite semigroups that we refer to as E-extended power graphs. An E-extended power graph ΓS of a finite semigroup S can be considered as the union of the power graph P(S) of S and the tree generated by E(S), where E(S) is the set of idempotents of S. Actually, ΓS is a minimal connected graph with vertex set S and containing P(S). In particular, if S is a finite semigroup with a unique idempotent we have ΓS=P(S). We give the number of spanning trees of an E-extended power graph of a finite semigroup, in particular, (Zn,⋅). Finally, as an application we discuss the number of spanning trees of an E-extended power graph of completely simple semigroups and completely regular semigroups.

Some new properties on N semigroups

In this study we first show that  satisfies two important homological properties, namely Rees short exact sequence an d short five lemma. In addition, by defining inversive semigroup varieties of  we prove that strictly inverse semigroup  is isomorphic to the spined product of (C)-inversive semigroup and the idempotent semigroup of . Moreover, we give some consequences of the results to make a detailed classification over . It has been recently defined a new semigroup  based on Rees matrix and completely 0-simple semigroups. Further, it has been also proved finiteness conditions and the existence of some fundamental properties over .

Emerging Theorems: An Amalgamative Outcome of Simple, 0-Simple, Semigroups, Principal factors an Expounding Assortment and Automation of semi-groups with space valued Fuzzy weakly inside principles

Before tackling the question we should perhaps begin by saying what a semigroup is. A non-empty set S endowed with a single binary operation is known as a semigroup if for evey x,y,z [2][3]in S, If in addition there exists 1 in S such that, for every x in S, we say that S is a semigroup with identity or (more usually) a monoid. In this article, we give a definition of an space valued fuzzy weakly inside ideal. We study some interesting properties [4] [5] [6] of space valued fuzzy weakly inside ideals and the relationship between space valued fuzzy weakly interior ideals and space valued fuzzy principles. We characterize some semigroups by using interval valued fuzzy weakly interior principles. Moreover, we found theorems of the homorphic image and the preimage of an space valued fuzzy weakly inside principle in semigroups [7]. The 0-simple semigroups begins with some elementary results on simple and 0-simple semigroups and a decomposition theorem for semigroups in general that indicates why and understanding of simple and 0-simple semigroups is important. The main result of the chapter is a structure theorem due to Rees (1940) [2, 31] which applies to 0-simple semigroups satisfying both the minimal conditions -- what are called completely 0-simple semigroups. Rees himself used a different definition, but we shall see below theorems that the two definitions are equivalent [1].

On the Inclusion Ideal Graph of Semigroups

The inclusion ideal graph [Formula: see text] of a semigroup [Formula: see text] is an undirected simple graph whose vertices are all the nontrivial left ideals of [Formula: see text] and two distinct left ideals [Formula: see text], [Formula: see text] are adjacent if and only if either [Formula: see text] or [Formula: see text]. The purpose of this paper is to study algebraic properties of the semigroup [Formula: see text] as well as graph theoretic properties of [Formula: see text]. We investigate the connectedness of [Formula: see text] and show that the diameter of [Formula: see text] is at most 3 if it is connected. We also obtain a necessary and sufficient condition of [Formula: see text] such that the clique number of [Formula: see text] is the number of minimal left ideals of [Formula: see text]. Further, various graph invariants of [Formula: see text], viz. perfectness, planarity, girth, etc., are discussed. For a completely simple semigroup [Formula: see text], we investigate properties of [Formula: see text] including its independence number and matching number. Finally, we obtain the automorphism group of [Formula: see text].

Finite Idempotent Set-Theoretic Solutions of the Yang–Baxter Equation

Abstract It is proven that finite idempotent left non-degenerate set-theoretic solutions $(X,r)$ of the Yang–Baxter equation on a set $X$ are determined by a left simple semigroup structure on $X$ (in particular, a finite union of isomorphic copies of a group) and some maps $q$ and $\varphi _{x}$ on $X$, for $x\in X$. This structure turns out to be a group precisely when the associated Yang–Baxter monoid $M(X,r)$ is cancellative and all the maps $\varphi _{x}$ are equal to an automorphism of this group. Equivalently, the Yang–Baxter algebra $K[M(X,r)]$ is right Noetherian, or in characteristic zero it has to be semiprime. The Yang–Baxter algebra is always a left Noetherian representable algebra of Gelfand–Kirillov dimension one. To prove these results, it is shown that the Yang–Baxter semigroup $S(X,r)$ has a decomposition in finitely many cancellative semigroups $S_{u}$ indexed by the diagonal, each $S_{u}$ has a group of quotients $G_{u}$ that is finite-by-(infinite cyclic) and the union of these groups carries the structure of a left simple semigroup. The case that $X$ equals the diagonal is fully described by a single permutation on $X$.

On the intersection ideal graph of semigroups

The intersection ideal graph Γ(S) of a semigroup S is a simple undirected graph whose vertices are all nontrivial left ideals of S and two distinct left ideals I, J are adjacent if and only if their intersection is nontrivial. In this paper, we investigate the connectedness of Γ(S). We show that if Γ(S) is connected, then the diameter of Γ(S) is at most two. Further, we classify the semigroups S in terms of their ideals such that the diameter of Γ(S) is two. We obtain the domination number, independence number, girth and the strong metric dimension of Γ(S). We have also investigated the completeness, planarity and perfectness of Γ(S). We show that if S is a completely simple semigroup, then Γ(S) is weakly perfect. More over, in this article, we give an upper bound of the chromatic number of Γ(S). Finally, if S is the union of n minimal left ideals, then we obtain the metric dimension and the automorphism group of Γ(S).

Канторовость квазиунитарных полигонов над вполне (0-)простыми полугруппами

An universal algebra $A$ is called cantorian if for any algebra $B$ of the same signature, the existence of injective homomorphisms $A\to B$ and $B \to A$ implies an isomorphism of algebras $A$ and $B$. A right act $X$ over a semigroup $S$ is called quasiunitary if $X=XS$. We prove that every quasiunitary act over a completely simple semigroup and also every quasiunitary act with zero over a completely 0-simple semigroup are cantorian.

Semigroup structures via IVI-octahedron sets

In this paper, we use IVI-octahedron ideals and IVI-octahedron bi-ideals to discuss some characterizations of special semigroups, e.g., an intra-regular semigroup, a semigroup which is semilattice of left [respectively, right] simple semigroups or simple left groups and a semigroup which is a semilattice of groups and a [respectively, left and right] simple semigroups. Finally, we define an IVIO-simple [IVIOL-simple and IVI) R-simple] semigroup and study some of its properties and we give a characterization of a semisimple semigroup by IVIOIs.

Acts Over Semigroups

We present a review of results obtained mainly in the last two decades in a number of areas of the theory of acts over semigroups. The authors limited themselves to the structural theory of acts. The acts over completely (0-)simple semigroups, with conditions on the congruence lattice, diagonal acts, biacts and multiacts, and also partial acts are considered. Our work is an expanded version of the report made by the authors in October 2017 at the conference of Institute of Mathematics of Technical University of Berlin, dedicated to the 75th anniversary of Professor Ulrich Knauer, supplemented by results of later works.

Cross-connection structure of locally inverse semigroups

Locally inverse semigroups are regular semigroups whose idempotents form pseudo-semilattices. We characterize the categories that correspond to locally inverse semigroups in the realm of Nambooripad’s cross-connection theory. Further, we specialize our cross-connection description of locally inverse semigroups to inverse semigroups and completely [Formula: see text]-simple semigroups, obtaining structure theorems for these classes. In particular, we show that the structure theorem for inverse semigroups can be obtained using only one category, quite analogous to the Ehresmann–Schein–Nambooripad Theorem; for completely [Formula: see text]-simple semigroups, we show that cross-connections coincide with structure matrices, thus recovering the Rees Theorem by categorical tools.

Analysis of semigroups with soft intersection ideals

Abstract In this paper, semisimple semigroups, duo semigroups, right (left) zero semigroups, right (left) simple semigroups, semilattice of left (right) simple semigroups, semilattice of left (right) groups and semilattice of groups are characterized in terms of soft intersection semigroups, soft intersection ideals of semigroups. Moreover, soft normal semigroups are defined and some characterizations of semigroups with soft normality are given.

New Generalizations of sup-Hesitant Fuzzy Ideals of Semigroups

As general concepts of sup-hesitant fuzzy right (resp., left, interior, two-sided) ideals of semigroups, the concepts of sup+α-hesitant fuzzy right (resp., left, interior, two-sided) ideals and sup-β-hesitant fuzzy right (resp., left, interior, two-sided) ideals are introduced and their properties are investigated. Then, the concepts are established by fuzzy sets, Łukasiewicz fuzzy sets, Łukasiewicz anti-fuzzy sets, Pythagorean fuzzy sets, hesitant fuzzy sets, hybrid sets, interval-valued fuzzy sets and cubic sets. Finally, we characterize which is intra-regular, completely regular, simple semigroups or another type of semigroups in terms of sup+α-type and sup-β-type of hesitant fuzzy sets.

Representations of Munn algebras and related semigroups

We establish representation types (finite, tame or wild) of finite dimensional Munn algebras with semisimple bases. As an application, we establish representation types of finite Rees matrix semigroups, in particular, 0-simple semigroups, and their mutually annihilating unions.

Regular congruence on a simple semigroup with a minimal right ideal

Regular congruence on a simple semigroup with a minimal right ideal

Knowledgebra: An Algebraic Learning Framework for Knowledge Graph

Knowledge graph (KG) representation learning aims to encode entities and relations into dense continuous vector spaces such that knowledge contained in a dataset could be consistently represented. Dense embeddings trained from KG datasets benefit a variety of downstream tasks such as KG completion and link prediction. However, existing KG embedding methods fell short to provide a systematic solution for the global consistency of knowledge representation. We developed a mathematical language for KG based on an observation of their inherent algebraic structure, which we termed as Knowledgebra. By analyzing five distinct algebraic properties, we proved that the semigroup is the most reasonable algebraic structure for the relation embedding of a general knowledge graph. We implemented an instantiation model, SemE, using simple matrix semigroups, which exhibits state-of-the-art performance on standard datasets. Moreover, we proposed a regularization-based method to integrate chain-like logic rules derived from human knowledge into embedding training, which further demonstrates the power of the developed language. As far as we know, by applying abstract algebra in statistical learning, this work develops the first formal language for general knowledge graphs, and also sheds light on the problem of neural-symbolic integration from an algebraic perspective.

A construction of left equalizer simple medial semigroups

A semigroup S is said to be left equalizer simple if, for all elements a, b, x, y of S, xa=xb implies ya=yb. It is known that left equalizer simplicity is a necessary condition for a semigroup to be embedded in a left simple semigroup. A semigroup satisfying the identity axyb=ayxb is called a medial semigroup. In this paper we show how to construct left equalizer simple medial semigroups, especially, medial semigroups which can be embedded in idempotent-free left simple semigroups.

A characterization of strong semilattices of periodic groups and rectangular bands by disjunctions of identities

Each completely regular semigroup is a semilattice of completely simple semigroups. The more specific concept of a strong semilattice provides the concrete product between two arbitrary elements. We characterize strong semilattices of rectangular groups by so-called disjunctions of identities. Disjunctions of identities generalize the classical concept of an identity and of a variety, respectively. The rectangular groups will be on the one hand left zero semigroups and right zero semigroups and on the other hand groups of exponent [Formula: see text], where [Formula: see text] is any set of pairwise coprime natural numbers.

(Completely) weak simple semigroups and (Completely) weak 0-simple semigroups

The structure theorems for (Completely) simple semigroups and (Completely) 0-simple semigroups have proved a powerful tool in the investigation of such semigroups. In this paper, first of all, we define weak simple semigroups and weak 0-simple semigroups and compare them with simple semigroups and 0-simple semigroups respectively. Then we give examples of these semigroups and describe the structure of them. Also, we define completely weak simple semigroup and completely weak 0-simple semigroup. Finally, by using Green’s equivalences, we prove some results and give equivalences, for these semigroups.