On the generation of synthetic features based on support chains and arbitrary metrics within a topological approach to data analysis. Part 1. Inclusion of empirical distance functions into the formalism

The analysis of the formalism of topological recognition theory based on the fundamental concepts of functional analysis made it possible to propose previously unexplored approaches to determining lattice estimates. In particular, the use of support chains for the analysis of Boolean lattices formed over Zhuravlev-regular sets of precedents has pointed to a new direction of research which consists in replacing estimates of lattice elements with certain types of functions and/or vectors. This extension of the formalism also allows for a systematic study of semiempirical distance functionals known in the literature to solve applied problems. Promising directions for further development of the formalism are substantiated including the functionals reducing descriptions of sets of a Boolean lattice to scalars and the development of a mathematical apparatus for the analysis of lattices where operations on the corresponding functions are involved. The latter direction is interesting as it allows defining lattice metrics without using lattice estimates.