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In this paper, we introduce a notion of dependencybetween subsets of an arbitrary fixed non-empty set Ω. To be more detailed, we introduce a preorder ←on the power set P(Ω)having the further property that B←Aif and only if {b}←Afor any b∈B. We shall argue that this relation generalizes well-studied notions of dependence occurring in such fields as linear algebra, topology, and combinatorics. Furthermore, we show that this relation is characterized by two set operators whose fixed points have interesting geometric and order-theoretic properties. After giving some some elementary results about such a dependency relation, we provide some specific examples taken from graph theory. An interesting property we will provide consists of the possibility to characterize partial orders on a finite lattice in terms of a suitable dependency relation. Finally, we introduce and analyze some specific classes of dependency relations, namely attractiveand anti-attractivedependency relations.

In this paper, we introduce a symmetry geometryfor 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 $\unicode[STIX]{x1D6FA}$be a given set. A pairing$\mathfrak{P}$on $\unicode[STIX]{x1D6FA}$is a triple $\mathfrak{P}:=(U,F,\unicode[STIX]{x1D6EC})$, where $U$and $\unicode[STIX]{x1D6EC}$are nonempty sets and $F:U\times \unicode[STIX]{x1D6FA}\rightarrow \unicode[STIX]{x1D6EC}$is a map having domain $U\times \unicode[STIX]{x1D6FA}$and codomain $\unicode[STIX]{x1D6EC}$. Through this notion, we introduce a local symmetry relation on $U$and a global symmetry relation on the power set ${\mathcal{P}}(\unicode[STIX]{x1D6FA})$. Based on these two relations, we establish the basic properties of our symmetry geometry induced by $\mathfrak{P}$. The basic tool of our study is a closure operator $M_{\mathfrak{P}}$, 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.

In this work we use the granular computing paradigm to study specific types of families of subsets, operators and families of ordered pairs of sets of attributes which are naturally induced by information tables. In an unifying perspective, by means of some representation results, we connect the study of finite closure systems, matroids and finite lattice theory in the scope of the more general notion of attribute dependency based on information tables. For a fixed finite set Ω and for a corresponding information table J having attribute set Ω, the fundamental tool we use to proceed in our investigation is the equivalence relation ≈ J on the power set P ( Ω ) which identifies two any sets of attributes inducing the same indiscernibility relation on the object set of J . We interpret the attribute dependency as a preorder ≥ J on P ( Ω ) whose induced equivalence relation coincides with ≈ J . Then we investigate in detail the links between the preorder ≥ J , a closure system and an abstract simplicial complex on Ω naturally induced by ≈ J and specific families of ordered pairs of sets of attributes on Ω. [ABSTRACT FROM AUTHOR]