Symmetry Geometry by Pairings

In this paper, we introduce a symmetry geometry for all those mathematical structures which can be characterized by means of a generalization (which we call pairing) of a finite rectangular table. In more detail, let Ω be a given set. A pairing P on Ω is a triple P := (U, F, Λ), where U and Λ are nonempty sets and F : U × Ω → Λ is a map having domain U × Ω and codomain Λ. Through this notion, we introduce a local symmetry relation on U and a global symmetry relation on the power set P(Ω). Based on these two relations, we establish the basic properties of our symmetry geometry induced by P. The basic tool of our study is a closure operator MP, by means of which (in the finite case) we can represent any closure operator. We relate the study of such a closure operator to several types of others set operators and set systems which refine the notion of an abstract simplicial complex