Pairings and related symmetry notions

Abstract

In this paper, we give a purely mathematical generalization of an information table. We call pairing on a given set $$\Omega $$ a triple $$\mathfrak {P}=(U, F, \Lambda )$$ , where U and $$\Lambda $$ are non-empty sets and $$F:U\times \Omega \rightarrow \Lambda $$ is a map. We provide several examples of pairings: graphs, digraphs, metric spaces, group actions and vector spaces endowed with a bilinear form. Moreover, we reinterpret the usual notion of indiscernibility (with respect to a fixed attribute subset of an information table) in terms of local symmetry on U and, then, we study a global version of symmetry, that we called indistinguishability. In particular, we interpret the latter relation as the symmetrization of a pre-order $$\le _{\mathfrak {P}}$$ , that describes the symmetry transmission between subsets of $$\Omega $$ . Hence, we introduce a global average of symmetry transmission and studied it for some basic digraph families. Finally, we prove that the partial order of any finite lattice can be described in terms of the above pre-order.