Fuzzy sets as texture spaces. I: Representation theorems
For \(S\) a set, if \({\mathcal L}\) is a completely distributive sublattice of the power set of \(S\) (with unions as suprema and with intersections as infima) which separates points of \(S\) (i.e., given \(s_1\neq s_2\) in \(S\), there exists \(L_i\in{\mathcal L}\) such that \(s_i\in L_i\) and \(s_j\not\in L_i\) for \(i,j= 1,2\), \(i\neq j\)), then \((S,{\mathcal L})\) is called a texture space. It has a complementation if there exists an inclusion-reversing involution on \({\mathcal L}\). Representation theorems for fuzzy sets, completely distributive lattices with an order-reversing involution, generalized fuzzy sets and intuitionistic sets by complemented texture spaces are presented.
scientific article; zbMATH DE number 1423468