Maximal objects and minimal objects in the sets with operations (Q2804139)

The notions of maximal open sets and maximal closed sets (dually those of minimal open sets and minimal closed sets), in topological spaces were introduced earlier by the authors. In this paper, the authors generalize some common properties of these concepts by imposing an axiom on a family \(\mathcal{S}\subset P(X)\), where \(P(X)\) is the power set of a set \(X\). Calling any set \(A\in \mathcal{S}\) an object of \(\mathcal{S}\) the authors define: A proper nonempty object \(A\in \mathcal{S}\) is called maximal object (minimal object) in \(\mathcal{S}\) if any object in \(\mathcal{S}\) which contains \(A\) (which is contained in \(A\)) is \(X\) or \(A\) (is \(\emptyset \) or \(A\)).NEWLINENEWLINEClosedness of \(\mathcal{S}\) under finite unions is used to generalize some results for maximal open sets as maximal objects in \(\mathcal{S}\) and closedness of \(\mathcal{S}\) under finite intersections is used to generalize some results for minimal open sets as minimal objects in \(\mathcal{S}\). Sets with operations \(\kappa\) on families of subsets are studied as examples, and some properties of maximal \(\kappa \)-open sets and minimal \(\kappa\)-closed sets are obtained.