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Similarities in structured spaces of sets

The objective of this paper is to present methodology for similarity evaluation of structured spaces of sets inspired by human cognitive processes. In contrast to classical similarity relations, which can operate only within the same space, our method can be applied to separate spaces. Proposed formulas are designed to compare two families of sets belonging to separate spaces. Unlike in set-theoretic approach to similarity present in literature, fundamental knowledge, which we use is sets and subsets cardinalities and division of spaces into subsets combined with appropriate minimum and maximum as aggregation operators. Theoretical discussion is supported with a case study, where we apply designed formulas to calculate similarities of four cities. Introduced method has been constructed after an analysis how humans perform similarity evaluation for hard to compare concepts and phenomena.