Structure Of Semigroups Research Articles

Endomorphism semigroups of 2-nilpotent binary relations

We define two constructions of semigroups, which up to isomorphism describe the structure of all endomorphism semigroups of 2-nilpotent binary relations.

A remark on the classifications of rhotrices as abstract structures

This paper presents additional classifications of rhotrices as abstract structures of Ring, Field, Integral Domain, Principal Ideal Domain and Unique Factorization Domain, as a sort of an additional work to our earlier classifications of rhotrices as algebraic structures of Groups, Semi groups, Monoids and Boolean Algebra. Rhotrix is a new paradigm of matrix theory, concerned with representing arrays of real numbers in mathematical rhomboid form, as an extension of ideas on matrix-tertions and matrix noitrets proposed by Atanassov and Shannon. Key words: Rhotrices, ring, field, integral domain, principal ideal domain, unique factorization domain.

Sensitivity to electrochemical potential of the number of off-resonant conduction states in a single-atom nanowire

Given a one-atom nanowire in the presence of an electrochemical potential, a novel formulation is presented so that the sensitivity to this potential of the number of conducting off-resonant channels is evaluated from defining a suitable parameter which can be viewed as a normalized gradient of the above number with respect to the electrochemical potential. The range of values taken on by this parameter is determined in terms of the electrochemical potential which is assumed to take on strictly positive values only. In fact, sufficiently large strictly positive values of that potential are considered. In addition, a linear operator associated with the sensitivity parameter is introduced and discussed within a commutative semigroup structure.

The Ideal Structure of Semigroups of Linear Transformations with Upper Bounds on Their Nullity or Defect

Suppose V is a vector space with dim V = p ≥ q ≥ ℵ0, and let T(V) denote the semigroup (under composition) of all linear transformations of V. For α ∈ T (V), let ker α and ran α denote the “kernel” and the “range” of α, and write n(α) = dim ker α and d(α) = codim ran α. In this article, we study the semigroups AM(p, q) = {α ∈ T(V):n(α) < q} and AE(p, q) = {α ∈ T(V):d(α) < q}. First, we determine whether they belong to the class of all semigroups whose sets of bi-ideals and quasi-ideals coincide. Then, for each semigroup, we describe its maximal regular subsemigroup, and we characterise its Green's relations and (two-sided) ideals. As a precursor to further work in this area,, we also determine all the maximal right simple subsemigroups of AM(p, q).

On the relationship between the rotation construction and ordered Abelian groups

By analogy with the usual extension of the group operation from the positive cone of an ordered Abelian group into the whole group, a construction—called symmetrization—is defined and it is related to the rotation construction [Jenei, On the structure of rotation-invariant semigroups, Archive for Mathematical Logic 42 (2003) 489–514]. Symmetrization turns out to be a kind of dualized rotation. A characterization is given for the left-continuous t-conorms for which their symmetrization is a uninorm. As a by-product a new family of involutive uninorms is introduced.

Associate inverse subsemigroups of regular semigroups

By an associate inverse subsemigroup of a regular semigroup S we mean a subsemigroup T of S containing a least associate of each x∈S, in relation to the natural partial order ≤ S . We describe the structure of a regular semigroup with an associate inverse subsemigroup, satisfying two natural conditions. As a particular application, we obtain the structure of regular semigroups with an associate subgroup with medial identity element.

Partial semigroups and primality indicators in the fractal generation of binomial coefficients to a prime square modulus

AbstractThis paper, resulting from two summer programs of Research Experience for Undergraduates, examines the congruence classes of binomial coefficients to a prime square modulus as given by a fractal generation process for lattice path counts. The process depends on the isomorphism of partial semigroup structures associated with each iteration. We also consider integrality properties of certain critical coefficients that arise in the generation process. Generalizing the application of these coefficients to arbitrary arguments, instead of just to the prime arguments appearing in their original function, it transpires that integrality of the coefficients is indicative of the primality of the argument.

Quantum families of maps and quantum semigroups on finite quantum spaces

Quantum families of maps between quantum spaces are defined and studied. We prove that quantum semigroup (and sometimes quantum group) structures arise naturally on such objects out of more fundamental properties. As particular cases we study quantum semigroups of maps preserving a fixed state and quantum commutants of given quantum families of maps.

Semigroups and their topologies arising from Green's left quasiorder

Given a semigroup (S, ·), Green’s left quasiorder on S is given by a ≤ b if a = u · b for some u ϵ S1. We determine which topological spaces with five or fewer elements arise as the specialization topology from Green’s left quasiorder for an appropriate semigroup structure on the set. In the process, we exhibit semigroup structures that yield general classes of finite topological spaces, as well as general classes of topological spaces which cannot be derived from semigroup structures via Green’s left quasiorder.

Bisimple monogenic orthodox semigroups

We give a complete description of the structure of all bisimple orthodox semigroups generated by two mutually inverse elements.

Zero-divisor semigroups and refinements of a star graph

Let G be a refinement of a star graph with center c . Let G c ∗ be the subgraph of G induced on the vertex set V ( G ) ∖ { c or end vertices adjacent to c } . In this paper, we completely determine the structure of commutative zero-divisor semigroups S whose zero-divisor graph G = Γ ( S ) and S satisfy one of the following properties: (1) G c ∗ has at least two connected components, (2) G c ∗ is a cycle graph C n of length n ≥ 5 , (3) G c ∗ is a path graph P n with n ≥ 6 , (4) S is nilpotent and Γ ( S ) is a finite or an infinite star graph. For any finite or infinite cardinal number n ≥ 2 , we prove that for any nilpotent semigroup S with zero element 0, S 4 = 0 if Γ ( S ) is a star graph K 1 , n . We prove that there is exactly one nilpotent semigroup S such that S 3 ≠ 0 and Γ ( S ) ≅ K 1 , n . For several classes of finite graphs G which are refinements of a star graph, we also obtain formulas to count the number of non-isomorphic corresponding semigroups.

The direct product of right zero semigroups and certain groupoids

This paper investigates first the structure of semigroups which are direct products of right zero semigroups and cancellative semigroups with identity. We consider the relationship of these semigroups to right groups (the direct products of groups and right zero semigroups). Finally, we consider groupoids which are direct products of right singular semigroups and unipotent (one-idempotent) groupoids with identity. Journal of the Nigerian Association of Mathematical Physics Vol. 10 2006: pp. 575-578

Further results in the theory of generalised inflations of semigroups

We determine the structure of semigroups that satisfy xyzw∈{xy,xw,zy,zw}. These semigroups are precisely those whose power semigroup is a generalised inflation of a band. The structure of generalised inflations of the following types of semigroups is determined: the direct product of a group and a band, a completely simple semigroup and a free semigroup F(X) on a set X. In the latter case the semigroup must be an inflation of F(X). We also prove that in any semigroup that equals its square, the power semigroup is a generalised inflation of a band if and only if it is an inflation of a band.

Wreath Product Structure of Left C-rpp Semigroups

The concept of wreath product of semigroups was first introduced by Neumann in 1960, and later on, this concept was used by Preston to investigate the structure of some inverse semigroups. In this paper, we modify the wreath product given by Neumann and Preston to study the structure of some generalized Clifford semigroups. In particular, we prove that a semigroup is a left C-rpp semigroup if and only if it is the wreath product of a left regular band and a C-rpp semigroup. Our result provides a new insight to the structure of left C-rpp semigroups.

Regularity properties in the classification program for separable amenable C*-algebras

We report on recent progress in the program to classify separable amenable C∗^*-algebras. Our emphasis is on the newly apparent role of regularity properties such as finite decomposition rank, strict comparison of positive elements, andZ\mathcal {Z}-stability, and on the importance of the Cuntz semigroup. We include a brief history of the program’s successes since 1989, a more detailed look at the Villadsen-type algebras which have so dramatically changed the landscape, and a collection of announcements on the structure and properties of the Cuntz semigroup.

Dividing Quantum Channels

We investigate the possibility of dividing quantum channels into concatenations of other channels, thereby studying the semigroup structure of the set of completely-positive trace-preserving maps. We show the existence of ‘indivisible’ channels which can not be written as non-trivial products of other channels and study the set of ‘infinitesimal divisible’ channels which are elements of continuous completely positive evolutions. For qubit channels we obtain a complete characterization of the sets of indivisible and infinitesimal divisible channels. Moreover, we identify those channels which are solutions of time-dependent master equations for both positive and completely positive evolutions. For arbitrary finite dimension we prove a representation theorem for elements of continuous completely positive evolutions based on new results on determinants of quantum channels and Markovian approximations.

On Countably Compact 0-Simple Topological Inverse Semigroups

We describe the structure of 0-simple countably compact topological inverse semigroups and the structure of congruence-free countably compact topological inverse semigroups.

On Closed Ideals of βG

For every discrete group G, the Stone-Cech compactification βG has a natural structure of a right topological semigroup. Using the concept of Boolean group ideal on G, we propose a way of constructing closed ideals of the semigroup βG. In particular, we show that, for every infinite Abelian group G of cardinality κ, there are \(2^{2^{\kappa}}\) distinct closed ideals in βG.

The semigroup generated by a unitary orbit of a singular matrix

We investigate the structure of the multiplicative semigroup generated by the set of matrices that are unitarily equivalent to a given singular matrix A. In particular, we give necessary and sufficient conditions, in terms of the singular values of A, for such a semigroup to consist of all matrices of rank not exceeding the rank of A.

Linearly ordered semigroups for fuzzy set theory

The fuzzy set theory initiated by Zadeh (Information Control 8:338---353, 1965) was based on the real unit interval [0,1] for support of membership functions with the natural product for intersection operation. This paper proposes to extend this definition by using the more general linearly ordered semigroup structure. As Moisil (Essais sur les Logiques non Chrysippiennes. Academie des Sciences de Roumanie, Bucarest, 1972, p. 162) proposed to define Lukasiewicz logics on an abelian ordered group for truth values set, we give a simple negative answer to the question on the possibility to build a Many-valued logic on a finite abelian ordered group. In a constructive way characteristic properties are step by step deduced from the corresponding set theory to the semigroup order structure. Some results of Clifford on topological semigroups (Clifford, A.H., Proc. Amer. Math. Soc. 9:682---687, 1958; Clifford, A.H., Trans. Amer. Math. Soc. 88:80---98, 1958), Paalman de Miranda work on I-semigroups (Paalman de Miranda, A.B., Topological Semigroups. Mathematical Centre Tracts, Amsterdam, 1964) and Schweitzer, Sklar on T-norms (Schweizer, B., Sklar, A., Publ. Math. Debrecen 10:69---81, 1963; Schweizer, B., Sklar, A., Pacific J. Math. 10:313---334, 1960; Schweizer, B., Sklar, A., Publ. Math. Debrecen 8:169---186, 1961) are revisited in this framework. As a simple consequence of Faucett theorems (Proc. Amer. Math. Soc. 6:741---747, 1955), we prove how canonical properties from the fuzzy set theory point of view lead to the Zadeh choice thus giving another proof of the representation theorem of T-norms. This structural approach shall give a new perspective to tackle the question of G. Moisil about the definition of discrete Many-valued logics as approximation of fuzzy continuous ones.