Search Results (12)

This paper aims to introduce a novel idea of possibility multi-fuzzy soft ordered semigroups for ideals and interior ideals. Various results, formulated as theorems based on these concepts, are presented and further validated with suitable examples. This paper also explores the broad applicability of possibility multi-fuzzy soft ordered semigroups in solving modern decision-making problems. Furthermore, this paper explores various classes of ordered semigroups, such as simple, regular, and intra-regular, using this innovative method. Based on these concepts, some important conclusions are drawn with supporting examples. Moreover, it defines the possibility of multi-fuzzy soft ideals for semiprime ordered semigroups. Full article

This paper considers a nonlinear impulsive fractional boundary value problem, which involves a $\psi$-Caputo-type fractional derivative and integral. Combining critical point theory and fractional calculus properties, such as the semigroup laws, and relationships between the fractional integration and differentiation, new multiplicity results of infinitely many solutions are established depending on some simple algebraic conditions. Finally, examples are also presented, which show that Caputo-type fractional models can be more accurate by selecting different kernels for the fractional integral and derivative. Full article

Consider a simple graph G with vertex set $V\left(G\right)$ and edge set $E\left(G\right)$. A graph invariant for G is a number related to the structure of G, which is invariant under the symmetry of G. The Sombor index of G is a new graph invariant defined as $SO\left(G\right)={\sum }_{uv\in E\left(G\right)}\sqrt{{\left(d_u\right)}^2+{\left(d_v\right)}^2}$. In this work, we connected the theory of the Sombor index with abstract algebra. We computed this topological index over the tensor and Cartesian products of a monogenic semigroup graph by presenting two different algorithms; the obtained results are illustrated by examples. Full article

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. Full article

A generalization of fractional vector calculus (FVC) as a self-consistent mathematical theory is proposed to take into account a general form of non-locality in kernels of fractional vector differential and integral operators. Self-consistency involves proving generalizations of all fundamental theorems of vector calculus for generalized kernels of operators. In the generalization of FVC from power-law nonlocality to the general form of nonlocality in space, we use the general fractional calculus (GFC) in the Luchko approach, which was published in 2021. This paper proposed the following: (I) Self-consistent definitions of general fractional differential vector operators: the regional and line general fractional gradients, the regional and surface general fractional curl operators, the general fractional divergence are proposed. (II) Self-consistent definitions of general fractional integral vector operators: the general fractional circulation, general fractional flux and general fractional volume integral are proposed. (III) The general fractional gradient, Green’s, Stokes’ and Gauss’s theorems as fundamental theorems of general fractional vector calculus are proved for simple and complex regions. The fundamental theorems (Gradient, Green, Stokes, Gauss theorems) of the proposed general FVC are proved for a wider class of domains, surfaces and curves. All these three parts allow us to state that we proposed a calculus, which is a general fractional vector calculus (General FVC). The difficulties and problems of defining general fractional integral and differential vector operators are discussed to the nonlocal case, caused by the violation of standard product rule (Leibniz rule), chain rule (rule of differentiation of function composition) and semigroup property. General FVC for orthogonal curvilinear coordinates, which includes general fractional vector operators for the spherical and cylindrical coordinates, is also proposed. Full article

We give a new approximation with respect of the traditional parametrix method of the solution of a parabolic equation whose generator is of big order and under the Hoermander form. This generalizes to a higher order generator the traditional approximation of Stratonovitch diffusion which put in relation random ordinary differential equation (the leading process is random and of finite energy. When a trajectory of it is chosen, the solution of the equation is defined) and stochastic differential equation (the leading process is random and only continuous and we cannot choose a path in the solution which is only almost surely defined). We consider simple operators where the computations can be fully performed. This approximation fits with the semi-group only and not for the full path measure in the case of a stochastic differential equation. Full article

The objective of this paper is put forward the novel concept of possibility fuzzy soft ideals and the possibility of fuzzy soft interior ideals. The various results in the form of the theorems with these notions are presented and further validated by suitable examples. In modern life decision-making problems, there is a wide applicability of the possibility fuzzy soft ordered semigroup which has also been constructed in the paper to solve the decision-making process. Elementary and fundamental concepts including regular, intra-regular and simple ordered semigroups in terms of possibility fuzzy soft ordered semigroup are presented. Later, the concept of left (resp. right) regular and left (resp. right) simple in terms of possibility fuzzy soft ordered semigroups are delivered. Finally, the notion of possibility fuzzy soft semiprime ideals in an ordered semigroup is defined and illustrated by theorems and example. Full article

We give a simple, straightforward proof of the non-hypercyclicity of an arbitrary (bounded or not) normal operator A in a complex Hilbert space as well as of the collection ${\left\{e^{tA}\right\}}_{t\ge 0}$ of its exponentials, which, under a certain condition on the spectrum of A, coincides with the $C_0$ -semigroup generated by it. We also establish non-hypercyclicity for symmetric operators. Full article

Let $P_2\subset P_1$ be a pair of weakly semiprime ideals of an ordered semigroup $(S,·,\le )$ . Then, the pair $P_2\subset P_1$ is called a weakly semiprime segment of S if ${\bigcap }_{n\in \mathbb{N}}{I}^n\subseteq P_2$ for all ideals I of S such that $P_2\subset I\subset P_1$ . In this paper, we classify weakly semiprime segments of an ordered semigroup into four types; those that are simple, exceptional, Archimedean, and decomposable. Full article

In this paper, we introduce and characterize the breakable semihypergroups, a natural generalization of breakable semigroups, defined by a simple property: every nonempty subset of them is a subsemihypergroup. Then, we present and discuss on an extended version of Rédei’s theorem for semi-symmetric breakable semihypergroups, proposing a different proof that improves also the theorem in the classical case of breakable semigroups. Full article

The semigroup D_V of digraphs on a set V of n labeled vertices is defined. It is shown that D_V is faithfully represented by the semigroup B_n of n ´ n Boolean matrices and that the Green’s L, R, H, and D equivalence classifications of digraphs in D_V follow directly from the Green’s classifications already established for B_n. The new results found from this are: (i) L, R, and H equivalent digraphs contain sets of vertices with identical neighborhoods which remain invariant under certain one-sided semigroup multiplications that transform one digraph into another within the same equivalence class, i.e., these digraphs exhibit Green’s isoneighborhood symmetries; and (ii) D equivalent digraphs are characterized by isomorphic inclusion lattices that are generated by their out-neighborhoods and which are preserved under certain two-sided semigroup multiplications that transform digraphs within the same D equivalence class, i.e., these digraphs are characterized by Green’s isolattice symmetries. As a simple illustrative example, the Green’s classification of all digraphs on two vertices is presented and the associated Green’s symmetries are identified. Full article