COMPARISON AMONG SEVERAL ADJACENCY PROPERTIES FOR A DIGITAL PRODUCT

Abstract

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 <TEX>$L_S$</TEX>- and the <TEX>$L_C$</TEX>-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 <TEX>$k_i$</TEX>-curves with <TEX>$l_i$</TEX> elements in <TEX>$Z^{n_i}$</TEX>, <TEX>$i{\in}\{1,2\}$</TEX> denoted by <TEX>$SC^{n_1,l_1}_{k_1}{\times}SC^{n_2,l_2}_{k_2}$</TEX>. Since a normal adjacency for this product and the <TEX>$L_C$</TEX>-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 <TEX>$Z^{n_1+n_2}$</TEX> and another adjacency satisfying the <TEX>$L_C$</TEX>-property? This research plays an important role in studying product properties of digital topological properties.