$\theta_f$-Approximations via fuzzy proximity relations: Semigroups in digital images

Abstract

This article introduces $\theta_{f}$-approximations of sets in fuzzy proximal relator space where $\theta\in\lbrack0,1)$. $\theta_{f}$-approximation provides a more sensitive approach for the upper approximations of subsets or subimages. $\theta_{f}$-approximation of a subimage are given with an example in digital images. Furthermore, $\theta_{f}$-approximately groupoid and semigroup in fuzzy proximal relator space are introduced.