Appl. Math. Inf. Sci. 8, No. 5, 2375-2387 (2014) 2375 Applied Mathematics & Information Sciences An International Journal http://dx.doi.org/10.12785/amis/080533 Simplicial Relative Cohomology Rings of Digital Images Ismet Karaca 1,∗ and Gulseli Burak 2 1 Department of Mathematics, Ege University, 35100 Izmir, Turkey 2 Department of Mathematics, Pamukkale University, 20070 Denizli, Turkey Received: 5 Sep. 2013, Revised: 3 Dec. 2013, Accepted: 4 Dec. 2013 Published online: 1 Sep. 2014 Abstract: The first goal of this paper is to show that the relative cohomology groups of digital images are determined algebraically by the relative homology groups of digital images. Then we state simplicial cup product for digital images and use it to establish ring structure of digital cohomology. Furthermore we give a method for computing the cohomology ring of digital images and give some examples concerning cohomology ring. Keywords: Digital simplicial relative cohomology group, cup product, cohomology ring. 1 Introduction In general calculating homology is not enought for determining differences between topological spaces. The cup product on cohomology is finer invariant. The cup product makes the cohomology group of a space into a ring. The ring structure from the cup product is an important advantage of cohomology theory over homology. While the homology groups of a space are equal to the cohomology groups, the ring structure on the cohomologies of the space is different. Then cup product can be used to distinguish the spaces. Cohomology groups are determined algebraically by the homology groups. We will define the relative cohomology groups of digital images and show that these satisfy basic properties very much like those for the relative homology of digital images. Althought basic properties of cohomology theory are similar to homology theory, there are some differences between them. One of the differences is that cohomology group is contravariant functor while homology group is covariant. Contravariance leads to additional structures in cohomology. These new structures are finer invariants of homotopy type and enable us to distinguish between topological spaces what are called cup products and cohomology operations. Many researchers(Rosenfeld [24], Kopperman [20], Kong [19], Malgouyres [21], Boxer [4, 5, 6, 7, 9], Han [11, 12], Karaca [1, 10]) have contributed to digital topology with their studies. They wish to characterize the properties of digital images with tools from Algebraic Topology. Their results play an important role in our study. Arslan, Karaca and Oztel [1], define simplicial homology group of a digital image and give examples of simplicial homology groups of certain digital images. They also compute simplicial homology groups of MSS 18 . Gonzalez-Diaz and Real [15] have their 14-adjacency algorithm to compute cup products on the simplicial complex. The advantage of this method is tried via a small program visualizing the several steps. Gonzalez-Diaz, Jimenez and Medrano [16] introduce a method for computing cup products on cubical approximations. Their cup products are computed directly from the cubical complex. Gonzalez-Diaz, Lamar and Umble [17] present how to simplify the combinatorial structure of cubical complex and obtain a homeomorphic cellular complex with fewer cells. They introduce formulas for a diagonal approximation on a general polygon and use it to compute cup products on the cohomology. The algorithm offered their work can be applied to compute cup products on any polyhedral ∗ Corresponding author e-mail: ismet.karaca@ege.edu.tr c  2014 NSP Natural Sciences Publishing Cor.