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In this article, we investigate the fundamental properties of coalgebras with coalgebra comultiplications, counits, and coalgebra homomorphisms of coalgebras over a commutative ring R with identity 1 R based on digital images with adjacency relations. We also investigate a contravariant functor from the category of digital images and digital continuous functions to the category of coalgebras and coalgebra homomorphisms based on digital images via the category of unitary R-modules and R-module homomorphisms. [ABSTRACT FROM AUTHOR]

In the current study, we explore digital homology modules, and investigate their fundamental properties on (pointed) digital images as one of the developments of symmetries. We also examine pointed digital Hopf spaces and base point preserving digital Hopf functions between pointed digital Hopf spaces with suitable digital multiplications, and explore the digital primitive homology classes, digital Pontryagin algebras on digital Hopf spaces as a symmetric phenomenon in mathematics and computer science. [ABSTRACT FROM AUTHOR]