Simplicial complexes and closure systems induced by indistinguishability relations

Lattice Representations with Set Partitions Induced by Pairings

“…Remark 4. There is another lattice which is order isomorphic to the dual closure lattice M(W), whose role is relevant when ones uses micro and macro granular representations induced by information tables (see [14] for details). Such a lattice is obtained by taking firstly the set G(W) := {[A] ≈ W : A ∈ M(W)}, and next by considering the partial order on G(W) defined by:…”

“…Thus, iterating the previous argument, we get F S (u, b) = F S (u , b). This shows the equivalence between (13) and (14). Therefore, in order to complete the proof of the theorem, fix an arbitrary element of u ∈ U S such that u ≡ A u − 1.…”

“…However, due to the fact that in the present paper the above sets U , Ω and Λ have only a formal nature, we will use the more mathematical term pairing instead of the previous ones. Hence, in our specific mathematical viewpoint, we can speak of algebraic GrC methods on pairings (for further results on such topics see also [13,14,17]).…”

“…In view of Property (4), we will use the terminology union additive relation induced by W to indicate the preorder W . As a type of dual structure of M(W), the subset family N(W) := {min([A] ≈ P ) : A ∈ M(W)} has been introduced in [14] showing that it is an abstract simplicial complex that is related with M(W) by means of several properties and results. For example, for some types of pairings M(W) and N(W) are respectively the closed subset family and the independent subset family of a matroid (see [14] for details).…”

“…Based on recent results obtained in the graph context [13,14] and on other general results linked to theoretical computer science [1,2,12,15,25,26,36,37,39], in this paper we continue the study of the basic property of the set system M(W) and use methods from GrC in relation to order-theory, limited to finite lattices.…”