Search

Let $X$ be a nonempty set and let $T(X)$ be the full transformation semigroup on $X$. The main objective of this paper is to study the subsemigroup $\overline{\Omega}(X, Y)$ of $T(X)$ defined by \[\overline{\Omega}(X, Y) = \{f\in T(X)\colon Yf = Y\},\] where $Y$ is a fixed nonempty subset of $X$. We describe regular elements in $\overline{\Omega}(X, Y)$ and show that $\overline{\Omega}(X, Y)$ is regular if and only if $Y$ is finite. We characterize unit-regular elements in $\overline{\Omega}(X, Y)$ and prove that $\overline{\Omega}(X, Y)$ is unit-regular if and only if $X$ is finite. We characterize Green's relations on $\overline{\Omega}(X, Y)$ and prove that $\mathcal{D} =\mathcal{J}$ on $\overline{\Omega}(X, Y)$ if and only if $Y$ is finite. We also determine ideals of $\overline{\Omega}(X, Y)$ and investigate its kernel. This paper extends several results appeared in the literature., Comment: 14

In this paper, we introduce and study a new kind of graph related to a unitary module [Formula: see text] on a commutative ring [Formula: see text] with identity, namely the annihilators comaximal graph of submodules of [Formula: see text], denoted by [Formula: see text]. The (undirected) graph [Formula: see text] is with vertices of all non-trivial submodules of [Formula: see text] and two vertices [Formula: see text] of [Formula: see text] are adjacent if and only if their annihilators are comaximal ideals of [Formula: see text], i.e. [Formula: see text]. The main purpose of this paper is to investigate the interplay between the graph-theoretic properties of [Formula: see text] and the module-theoretic properties of [Formula: see text]. We study the annihilators comaximal graph [Formula: see text] in terms of the powers of the decomposition of [Formula: see text] to product distinct prime numbers in some special cases.

Let [Formula: see text] be a pair of groups, in which [Formula: see text] is a normal subgroup of [Formula: see text] and [Formula: see text] is a free presentation of [Formula: see text]. Suppose that [Formula: see text] admits a complement in [Formula: see text], Ellis defined the notion of the Schur multiplier of the pair [Formula: see text] as [Formula: see text], where [Formula: see text] is a normal subgroup in a free group [Formula: see text] such that [Formula: see text]. In this paper, we give some inequalities for the order of the Schur multiplier of a pair of groups.

Let [Formula: see text] be a commutative ring with nonzero identity and [Formula: see text] be the set of all nonzero zero-divisors of [Formula: see text]. The zero-divisor graph of [Formula: see text], denoted by [Formula: see text], is a simple graph whose vertex set consists of all elements of [Formula: see text], and two distinct vertices [Formula: see text] and [Formula: see text] are adjacent if and only if [Formula: see text]. In this paper, we investigate the [Formula: see text]-spectral of the zero-divisor graph of the ring [Formula: see text]. Specially, we study the [Formula: see text]-spectral of the zero-divisor graph of the ring [Formula: see text] for [Formula: see text], [Formula: see text], [Formula: see text], where [Formula: see text], [Formula: see text] and [Formula: see text] are distinct prime numbers and [Formula: see text]. Moreover, we study the [Formula: see text]-spectral of the zero-divisor graph of the ring [Formula: see text], where [Formula: see text] is a prime number and [Formula: see text].

Any positive integer [Formula: see text] can be written uniquely as [Formula: see text] where [Formula: see text] is square-full, [Formula: see text] is square-free and [Formula: see text] We denote the square-full part and the square-free part of [Formula: see text] by [Formula: see text] and [Formula: see text] respectively. In this paper, we use an elementary method to study the distribution of square-free and square-full part of an integer. Moreover, we also study the distribution of primitive roots modulo a prime [Formula: see text] that are square-full number.

Consider a finite ordered set [Formula: see text] with a zig-zag order. In this paper, we consider the semigroup [Formula: see text] of all compressing-preserving transformations on [Formula: see text]. Necessary and sufficient conditions for [Formula: see text] to be regular and coregular are given. Regular and coregular elements in [Formula: see text] are completely described. For a subsemigroup [Formula: see text] of the transformation semigroup [Formula: see text], we obtain that left regular and right regular elements in [Formula: see text] are identical. A characterization of left [right] regular elements in [Formula: see text] is given. We classify all left [right] regular semigroups [Formula: see text].

The sum-connectivity index of a graph [Formula: see text] is defined as the sum of weights [Formula: see text] over all edges [Formula: see text] of [Formula: see text], where [Formula: see text] and [Formula: see text] are the degrees of the vertices [Formula: see text] and [Formula: see text] in [Formula: see text], respectively. In this paper, we provide a simpler method to the extremal trees and unicyclic graph of the sum-connectivity index.

In this paper, we introduce the new spaces [Formula: see text] of double sequences of bounded variation, where [Formula: see text], and we investigate some topological properties. Moreover, we prove some strict inclusion relations and we examine [Formula: see text]-dual of the spaces [Formula: see text]. Finally, we characterize some classes of four-dimensional matrices [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text], where [Formula: see text], [Formula: see text] and [Formula: see text].

This paper introduces the [Formula: see text]-fractal Sobolev system of functions corresponding to Sobolev orthonormal system of functions. An approximation-related result similar to Weierstrass theorem is derived. It is shown that the set of [Formula: see text]-fractal versions of Sobolev sums is dense and complete in the weighted Sobolev space [Formula: see text]. A Schauder basis and a Riesz basis of fractal type for the space [Formula: see text] are found. The Fourier–Sobolev expansion of an [Formula: see text]-fractal function [Formula: see text] corresponding to a certain set of interpolation points is presented. Moreover, some results on convergence of Fourier–Sobolev expansion of [Formula: see text] with respect to uniform norm and Sobolev norm are established.

Let [Formula: see text] and [Formula: see text] be two Banach algebras such that [Formula: see text] is a Banach [Formula: see text]-bimodule with the left and right compatible actions of [Formula: see text] on [Formula: see text]. Let [Formula: see text] be a strongly splitting Banach algebra extension of [Formula: see text] by [Formula: see text]. In this paper, we investigate some homological aspects such as injectivity, projectivity and flatness of [Formula: see text] and give some necessary and sufficient conditions for injectivity, projectivity and flatness of [Formula: see text].

Let [Formula: see text] be a graph with [Formula: see text] vertices and [Formula: see text] is the degree of its [Formula: see text]th vertex ([Formula: see text] is the degree of [Formula: see text]), then the Zagreb matrix of [Formula: see text] is the square matrix of order [Formula: see text] whose [Formula: see text]entry is equal to [Formula: see text] if the [Formula: see text]th and [Formula: see text]th vertex of [Formula: see text] are adjacent, and zero otherwise. The Zagreb energy [Formula: see text] of a graph [Formula: see text] is defined as the sum of the absolute values of the eigenvalues of the Zagreb matrix. In this paper, we compute the Zagreb energy for some of the specific graphs, edge deleted graphs and complements graphs. Moreover, some properties of the eigenvalues and bounds for Zagreb energy are also discussed.

Let [Formula: see text] be a graph of order [Formula: see text] and [Formula: see text] be the degree of the vertex [Formula: see text], for [Formula: see text]. The [Formula: see text] matrix of [Formula: see text] is the square matrix of order [Formula: see text] whose [Formula: see text]-entry is equal to [Formula: see text] if [Formula: see text] is adjacent to [Formula: see text], and zero otherwise. Let [Formula: see text], be the eigenvalues of [Formula: see text] matrix. The [Formula: see text] energy of a graph [Formula: see text], denoted by [Formula: see text], is defined as the sum of the absolute values of the eigenvalues of [Formula: see text] matrix. In this paper, we prove that the star has the minimum [Formula: see text] energy among trees.

The aim of this paper is to define the operator of [Formula: see text]-derivative based upon the Borel distribution and by using this operator, we familiarize a new subclass of [Formula: see text]-uniformly starlike functions [Formula: see text]-[Formula: see text] Further, we obtain coefficient estimates, distortion theorems, convex linear combinations and radii of close-to-convexity, starlikeness and convexity for functions [Formula: see text]-[Formula: see text] We also determine the second Hankel inequality for functions belonging to this subclass.

Let [Formula: see text] be a graph with edge set [Formula: see text]. For an edge [Formula: see text] in [Formula: see text], we define [Formula: see text], where [Formula: see text] and [Formula: see text] are degrees of vertices [Formula: see text] and [Formula: see text] in [Formula: see text], respectively. For [Formula: see text], the graph invariants [Formula: see text], [Formula: see text] and [Formula: see text] are defined as [Formula: see text], [Formula: see text] and [Formula: see text], where [Formula: see text] means that the edges [Formula: see text] and [Formula: see text] are incident. In this paper, some relationship between these graph invariants and some classical topological indices were presented. Moreover, some bounds for [Formula: see text], [Formula: see text] and [Formula: see text] are obtained and trees with the first through the third smallest [Formula: see text] and [Formula: see text], as well as the trees with the first through the forth smallest [Formula: see text] are also characterized.

Let [Formula: see text] be a lattice, [Formula: see text] an ideal of [Formula: see text] and [Formula: see text] the set of all elements of [Formula: see text] which are not prime to [Formula: see text]. In this paper, we investigate some interesting properties of [Formula: see text] and introduce the total graph of a lattice [Formula: see text] with respect to an ideal [Formula: see text]. It is the graph with all elements of [Formula: see text] as vertices and for distinct [Formula: see text], the vertices [Formula: see text] and [Formula: see text] are adjacent if and only if [Formula: see text]. We investigated the basic properties of this graph and its induced subgraphs as diameter, girth, clique number, cut vertex and independence number.

In this paper, we prove that if [Formula: see text] is a partitioning ideal of an [Formula: see text]-semiring [Formula: see text] with respect to two subsets [Formula: see text] and [Formula: see text] of [Formula: see text], then the quotient [Formula: see text]-semirings [Formula: see text] and [Formula: see text] are isomorphic. Also, the concepts of maximal homomorphism, steady homomorphism, one-one homomorphism and [Formula: see text]-homomorphism are generalized for [Formula: see text]-semirings and obtained relation between them. Finally, the fundamental theorem of semiring homomorphism is generalized for [Formula: see text]-semirings.

The irregularity of a simple undirected graph G is defined as irr(G) = Sigma(uv is an element of E(G)) vertical bar d(G)(u) - d(G) (v) vertical bar, where d(G) (u) denotes the degree of a vertex u is an element of V (G). The total irregularity of a graph as a new measure of graph irregularity is defined as irr(t) (G) = 1/2 Sigma(u,v is an element of V (G) )vertical bar d(G) (u) - d(G) (v) vertical bar. In this paper, the irregularity and the total irregularity of fractal graphs and the derived graphs from a class of fractal graphs are investigated. Exact formulae are presented for the computation of the irregularity and total irregularity of fractal-type graphs in terms of the parameters of the underlying graphs.

This paper studies the bi-compact-[Formula: see text]-open topology on the family [Formula: see text] of all continuous functions from a completely regular space [Formula: see text] to the space [Formula: see text]. We compare this topology with open-point topology, bi-point-open topology and bi-compact-open topology and mention an example of a space asked in [P. Garg and S. Kundu, The compact-[Formula: see text]-open topology on [Formula: see text], Topology Appl. 159(8) (2012) 2082–2089] which satisfies [Formula: see text]. We also investigate submertrizability, mertrizability, separability, [Formula: see text]ech-completeness and Baire property of bi-compact-[Formula: see text]-open topology on [Formula: see text].

The Young type inequality asserts that if [Formula: see text] are two positive real numbers, then for each [Formula: see text], we have [Formula: see text] In this paper, we obtain some new inequalities using two different weights. For example, if [Formula: see text], then [Formula: see text] [Formula: see text] where [Formula: see text] In addition, we refine some matrix inequalities for Unitarily invariant norms by applying the deduced numerical inequalities.

In this paper, we define a module [Formula: see text] to be [Formula: see text] if and only if intersection of each pair of [Formula: see text]-closed direct summands is also a direct summand of [Formula: see text]. We investigate structural properties of [Formula: see text]-modules and locate the implications between the other module properties which are essentially based on direct summands. We deal with decomposition theory as well as direct summands of [Formula: see text]-modules. We apply our results to matrix rings. To this end, it is obtained that the [Formula: see text] property is not Morita invariant.

In this paper, we introduce Fibonacci backward difference operator [Formula: see text] of fractional order l by the composition of Fibonacci band matrix [Formula: see text] and difference operator [Formula: see text] of fractional order l, defined by [Formula: see text] and introduce sequence spaces [Formula: see text] and [Formula: see text] We present some topological properties, obtain Schauder basis and determine [Formula: see text]-, [Formula: see text]- and [Formula: see text]-duals of the spaces [Formula: see text] and [Formula: see text] We characterize certain classes of matrix mappings from the spaces [Formula: see text] and [Formula: see text] to any of the space [Formula: see text] [Formula: see text] [Formula: see text] or [Formula: see text] Finally we compute necessary and sufficient conditions for a matrix operator to be compact on the spaces [Formula: see text] and [Formula: see text]