On the Digital Cohomology Modules

Abstract

In this paper, we consider the digital cohomology modules of a digital image consisting of a bounded and finite subset of Zn and an adjacency relation. We construct a contravariant functor from the category of digital images and digital continuous functions to the category of unitary R-modules and R-module homomorphisms via the category of cochain complexes of R-modules and cochain maps, where R is a commutative ring with identity 1R. We also examine the digital primitive cohomology classes based on digital images and find the relationship between R-module homomorphisms of digital cohomology modules induced by the digital convolutions and digital continuous functions.

Coalgebras on Digital Images

“…Digital topology was first studied in the late 1960s by A. Rosenfeld, and digital surfaces (or digital manifolds) were also developed in the early 1980s and in 1990s by many authors. In particular, the formal and informal definitions of a lot of terms in homotopy and simplicial (co)homology theory based on a digital image on Z 2 or Z 3 with adjacency relations were nicely described in [33][34][35][36][37][38]; see also [22] for digital quasi co-Hopf space, and [39,40] for digital cohomology modules and cone metric spaces.…”

“…It can be seen in [39] that, for each digital image (X, k X ), dH n (X; R) has the R-module structure whose scalar multiplications…”

“…Indeed, as a quotient R-module, dH n (X; R) has the unitary R-module structure because R is a commutative ring with identity; see [39] [Theorem 1] for further details.…”

“…Let D be the category of digital images and digital continuous functions as mentioned earlier in Remark 1, and let Mod R be the category of unitary R-modules and R-module homomorphisms. Then, it can be seen in[39] [Theorem 1] that the assignment dH…”

Splitting Extensions of Nonassociative Algebras and Modules with Metagroup Relations

“…In these works, it is shown that generalized Cayley-Dickson algebras are particular cases of metagroup algebras. For related with them digital Hopf spaces cohomologies were investigated in [24].…”