2022/11/09 by Kozin, I.V., Narzullaev, U. Kh.
paper · doi:10.34920/ijtaidt/vol_2022_issue_1_1
The article considers the extension of such concepts as a ball, a segment, a convex set to an arbitrary metric space. In particular, along with the Euclidean metric, the Manhattan metric and the supremal metric in a multidimensional vector space are considered. The Hamming metric in binary space and a number of metrics in the space of permutations are also considered. Segments are formally described in the binary space with the Hamming metric and in the space of permutations with the Kendall metric. It is shown that the interval in the Hamming metric coincides with the concept of a skhema (shim), which is used in the theory of genetic algorithms in the Holland model. It is proved that the segment between two permutations in the space of permutations with the Kendall metric consists of all permutations that preserve the relative orders induced by these permutations. The concept of structures with heredity is introduced, examples and properties of these structures are described. It is shown that the sets of segments in a metric space form a structure with heredity.