A generalization of Matkowski’s and Suzuki’s fixed point theorems

In this paper, we present a generalization of Suzuki’s fixed point theorem and the Matkowski contraction principle for a system of transformations on the finite product of metric spaces.

Infra Soft Semiopen Sets and Infra Soft Semicontinuity

To contribute to the area of infra soft topology, we introduce one of the generalizations of infra soft open sets called infra soft semiopen sets. We establish some characterizations of them and study their main properties. We determine under what condition this class is closed under finite intersection and show that this class is preserved under infra soft continuous mappings and finite product of soft spaces. Then, we present the concepts of infra semi-interior, infra semiclosure, infra semilimit, and infra semiboundary soft points of a soft set and elucidate the relationships between them. Finally, we exploit infra soft semiopen and infra soft semiclosed sets to define new types of soft mappings. We characterize each one of these soft mappings and explore main features.

Two Classes of Infrasoft Separation Axioms

One of the considerable topics in the soft setting is the study of soft topology which has enticed the attention of many researchers. To contribute to this scope, we devote this work to investigate two classes of separation axioms with respect to the distinct ordinary elements through one of the generalizations of soft topology called infrasoft topology. We first formulate the concepts of infra- t p -soft T j using total belong and partial nonbelong relations and then introduce the concepts of infra- t t -soft T j -spaces using total belong and partial nonbelong relations. To illustrate the relationships between them, we provide some examples. We discuss their fundamental properties and study their behaviors under some special types of infrasoft topologies. An extensive discussion is given for the transmission of these two classes between infrasoft topology and its parametric infratopologies. In the end, we demonstrate which ones have topological and hereditary properties, and we show their behaviors under the finite product of soft spaces.

Complete Hausdorffness and Complete Regularity on Supra Topological Spaces

The supra topological topic is of great importance in preserving some topological properties under conditions weaker than topology and constructing a suitable framework to describe many real-life problems. Herein, we introduce the version of complete Hausdorffness and complete regularity on supra topological spaces and discuss their fundamental properties. We show the relationships between them with the help of examples. In general, we study them in terms of hereditary and topological properties and prove that they are closed under the finite product space. One of the issues we are interested in is showing the easiness and diversity of constructing examples that satisfy supra T i spaces compared with their counterparts on general topology.

Closed sets of finitary functions between products of finite fields of coprime order

We investigate the finitary functions from a finite product of finite fields $$\prod _{j =1}^m\mathbb {F}_{q_j} = {\mathbb K}$$ ∏ j = 1 m F q j = K to a finite product of finite fields $$\prod _{i =1}^n\mathbb {F}_{p_i} = {\mathbb {F}}$$ ∏ i = 1 n F p i = F , where $$|{\mathbb K}|$$ | K | and $$|{\mathbb {F}}|$$ | F | are coprime. An $$({\mathbb {F}},{\mathbb K})$$ ( F , K ) -linearly closed clonoid is a subset of these functions which is closed under composition from the right and from the left with linear mappings. We give a characterization of these subsets of functions through the $${\mathbb {F}}_p[{\mathbb K}^{\times }]$$ F p [ K × ] -submodules of $$\mathbb {F}_p^{{\mathbb K}}$$ F p K , where $${\mathbb K}^{\times }$$ K × is the multiplicative monoid of $${\mathbb K}= \prod _{i=1}^m {\mathbb {F}}_{q_i}$$ K = ∏ i = 1 m F q i . Furthermore we prove that each of these subsets of functions is generated by a set of unary functions and we provide an upper bound for the number of distinct $$({\mathbb {F}},{\mathbb K})$$ ( F , K ) -linearly closed clonoids.

Periodic matrix difference equations and companion matrices in blocks: some applications

This study is devoted to some periodic matrix difference equations, through their associated product of companion matrices in blocks. Linear recursive sequences in the algebra of square matrices in blocks and the generalized Cayley–Hamilton theorem are considered for working out some results about the powers of matrices in blocks. Two algorithms for computing the finite product of periodic companion matrices in blocks are built. Illustrative examples and applications are considered to demonstrate the effectiveness of our approach.

Connectedness and Local Connectedness on Infra Soft Topological Spaces

This year, we introduced the concept of infra soft topology as a new generalization of soft topology. To complete our analysis of this space, we devote this paper to presenting the concepts of infra soft connected and infra soft locally connected spaces. We provide some descriptions for infra soft connectedness and elucidate that there is no relationship between an infra soft topological space and its parametric infra topological spaces with respect to the property of infra soft connectedness. We discuss the behaviors of infra soft connected and infra soft locally connected spaces under infra soft homeomorphism maps and a finite product of soft spaces. We complete this manuscript by defining a component of a soft point and establishing its main properties. We determine the conditions under which the number of components is finite or countable, and we discuss under what conditions the infra soft connected subsets are components.

Infra Soft Compact Spaces and Application to Fixed Point Theorem

Infra soft topology is one of the recent generalizations of soft topology which is closed under finite intersection. Herein, we contribute to this structure by presenting two kinds of soft covering properties, namely, infra soft compact and infra soft Lindelöf spaces. We describe them using a family of infra soft closed sets and display their main properties. With the assistance of examples, we mention some classical topological properties that are invalid in the frame of infra soft topology and determine under which condition they are valid. We focus on studying the “transmission” of these concepts between infra soft topology and classical infra topology which helps us to discover the behaviours of these concepts in infra soft topology using their counterparts in classical infra topology and vice versa. Among the obtained results, these concepts are closed under infra soft homeomorphisms and finite product of soft spaces. Finally, we introduce the concept of fixed soft points and reveal main characterizations, especially those induced from infra soft compact spaces.

Quasi-dual Baer modules

Let R be a ring and let M be an R-module with $$S={\text {End}}_R(M)$$ S = End R ( M ) . The module M is called quasi-dual Baer if for every fully invariant submodule N of M, $$\{\phi \in S \mid Im\phi \subseteq N\} = eS$$ { ϕ ∈ S ∣ I m ϕ ⊆ N } = e S for some idempotent e in S. We show that M is quasi-dual Baer if and only if $$\sum _{\varphi \in \mathfrak {I}} \varphi (M)$$ ∑ φ ∈ I φ ( M ) is a direct summand of M for every left ideal $$\mathfrak {I}$$ I of S. The R-module $$R_R$$ R R is quasi-dual Baer if and only if R is a finite product of simple rings. Other characterizations of quasi-dual Baer modules are obtained. Examples which delineate the concepts and results are provided.