Multiplicative Property of the Digital Fundamental Group

Abstract

Recent papers have partially discussed the multiplicative or the non-multiplicative property of the digital fundamental group. Thus, the paper studies a condition of which the multiplicative property of the digital fundamental group holds. Precisely, for two digital spaces with ki-adjacencies of \(\mathbf{Z}^{n_{i}}\) , denoted by (Xi,ki), i∈{1,2}, using the LHS- or LHC-property of the digital product (or Cartesian product of digital spaces) with k-adjacency (X1×X2,k), a k-homotopic thinning of the digital product, and various properties from digital covering and digital homotopy theories, we provide a method of calculating the k-fundamental group of the digital product. Furthermore, the notion of HT-(k0,k1)-isomorphism is established and used in calculating the k-fundamental group of a digital product. Finally, we find a condition of which the multiplicative property of the digital fundamental group holds. This property can be used in classifying digital spaces from the view points of digital homotopy theory, mathematical morphology, and digital geometry.