The k-homotopic thinning and a torus-like digital image in Zn supercript stop

In order to discuss digital topological properties of a digital image ( X, k), many recent papers have used the digital fundamental group and several digital topological invariants such as the k-linking number, the k-topological number, and so forth. Owing to some difficulties of an establishment of the multiplicative property of the digital fundamental group, a k-homotopic thinning method can be essentially used in calculating the digital fundamental group of a digital product with k-adjacency. More precisely, let SC(ki)(ni,li) be a simple closed k(i)-curve with l(i) elements in Z(ni), i epsilon {1,2}. For some k-adjacency of the digital product Sc(k1)(n1,l1) x Sc(k2)(n2,l2) subset of Z(n1 + n2) which is a torus-like set, proceeding with the k-homotopic thinning of Sc(k1)(n1,l1) x Sc(k2)(n2,l2) ,we obtain its k-homotopic thinning set denoted by DT(k). Writing an algorithm for calculating the digital fundamental group of Sc(k1)(n1,l1) x Sc(k2)(n2,l2) ,we investigate the k-fundamental group of (Sc(k1)(n1,l1) x Sc(k2)(n2,l2), k) by the use of various properties of a digital covering (Zx Z,p(1) x p(2),DT(k)), a strong k-deformation retract, and algebraic topological tools. Finally, we find the pseudo-multiplicative property ( contrary to the multiplicative property) of the digital fundamental group. This property can be used in classifying digital images from the view points of both digital k-homotopy theory and mathematical morphology.