American Institute of Mathematical Sciences

In this paper, we construct the smash product from the digital viewpoint and prove some its properties such as associativity, distributivity, and commutativity. Moreover, we present the digital suspension and the digital cone for an arbitrary digital image and give some examples of these new concepts.

Motivated by some ordinary and extreme connectedness properties of topologies, we introduce several reasonable connectedness properties of relators (families of relations). Moreover, we establish some intimate connections among these properties.

More concretely, we investigate relationships among various minimalness (well-chainedness), connectedness, hyper- and ultra-connectedness, door, superset, submaximality and resolvability properties of relators.

Since most generalized topologies and all proper stacks (ascending systems) can be derived from preorder relators, the results obtained greatly extends some standard results on topologies. Moreover, they are also closely related to some former results on well-chained and connected uniformities.

In this paper, we prove the existence of best proximity point and coupled best proximity point on metric spaces with partial order for weak proximal contraction mappings such that these critical points satisfy some constraint inequalities.

Let \begin{document}$ p $\end{document} be a fixed odd prime. The Bockstein free part of the mod \begin{document}$ p $\end{document} Steenrod algebra, \begin{document}$ \mathcal{A}_p $\end{document}, can be defined as the quotient of the mod \begin{document}$ p $\end{document} reduction of the Leibniz Hopf algebra, \begin{document}$ \mathcal{F}_p $\end{document}. We study the Hopf algebra epimorphism \begin{document}$ \pi\colon \mathcal{F}_p\to \mathcal{A}_p $\end{document} to investigate the canonical Hopf algebra conjugation in \begin{document}$ \mathcal{A}_p $\end{document} together with the conjugation operation in \begin{document}$ \mathcal{F}_p $\end{document}. We also give a result about conjugation invariants in the mod 2 dual Leibniz Hopf algebra using its multiplicative algebra structure.

In this paper, we introduce the concept of rough semi-uniform spaces as a supercategory of rough pseudometric spaces and approximation spaces. A completion of approximation spaces has been constructed using rough semi-uniform spaces. Applications of rough semi-uniform spaces in the construction of proximities of digital images is also discussed.

In this paper, we give the constructions of the coequalizer and coproduct objects for the category of crossed modules, in a modified category of interest (MCI). In other words, we prove that the corresponding category is finitely cocomplete.