Comparison Among Several Adjacency Properties for a Digital Product

Owing to the notion of a normal adjacency for a digital product in [8], the study of product properties of digital topological properties has been substantially done. To explain a normal adjacency of a digital product more efficiently, the recent paper [22] proposed an S-compatible adjacency of a digital product. Using an S-compatible adjacency of a digital product, we also study product properties of digital topological properties, which improves the presentations of a normal adjacency of a digital product in [8]. Besides, the paper [16] studied the product property of two digital covering maps in terms of the LS - and the LC -property of a digital product which plays an important role in studying digital covering and digital homotopy theory. Further, by using HS- and HC-properties of digital products, the paper [18] studied multiplicative properties of a digital fundamental group. The present paper compares among several kinds of adjacency relations for digital products and proposes their own merits and further, deals with the problem: consider a Cartesian product of two simple closed kᵢ -curves with lᵢ elements in Znᵢ , i∈\1,2\ denoted by SCn₁,l₁k₁×SCn₂,l₂k₂ . Since a normal adjacency for this product and the LC -property are different from each other, the present paper address the problem: for the digital product does it have both a normal k-adjacency of Zn₁+n₂ and another adjacency satisfying the LC -property? This research plays an important role in studying product properties of digital topological properties.