AAECC (2019) 30:299–311 https://doi.org/10.1007/s00200-018-0374-y ORIGINAL PAPER Approximately -semigroups in proximal relator spaces Mustafa Uçkun 1 · Ebubekir ˙ Inan 1 Received: 26 September 2018 / Revised: 29 October 2018 / Accepted: 30 October 2018 / Published online: 10 November 2018 © Springer-Verlag GmbH Germany, part of Springer Nature 2018 Abstract This article introduces approximately Γ -semigroups, approximately Γ -ideals and approximately upper Γ -ideals in proximal relator spaces. Some properties of descrip- tively approximations are investigated using by object description Γ -homomorphism. Furthermore, an example of approximately Γ -semigroup with working on digital images. Keywords Proximity spaces · Relator spaces · Descriptive approximations · Approximately semigroups Mathematics Subject Classification 08A05 · 68Q32 · 54E05 1 Introduction In proximal relator space, the elements of sets are considered non-abstract points. Non-abstract points describable with feature vectors such as digital images in proximal relator spaces. Descriptively upper approximation of a nonempty set is obtained by using the set of points composed by the proximal relator space together with matching features of points. In the algebraic structures constructed on proximal relator spaces, the basic tool is consideration of descriptively upper approximations of the subsets of non-abstract points. In a groupoid A in proximal relator space, the binary operation “◦” may be closed in descriptively upper approximation of A that is a ◦ b is in descriptively upper approximation of A for all a, b in A. Essentially, our aim is to obtain approximately Γ -semigroups, approximately Γ -ideals and approximately upper Γ -ideals in proximal relator spaces and some prop- erties of descriptively approximations using object description Γ -homomorphism. B Mustafa Uçkun muckun@adiyaman.edu.tr Ebubekir ˙ Inan einan@adiyaman.edu.tr 1 Department of Mathematics, Faculty of Arts and Sciences, Adıyaman University, 02040 Adıyaman, Turkey 123