Lcloseds

Lcloseds are a generalized class of subsets defined with respect to a closure-like operator L on a set X. Here, L is a map from the power set of X to itself that is monotone, and typically idempotent and extensive, so it behaves like a closure operator in abstract lattice terms. A subset A of X is called an lclosed (or Lclosed) set if it is a fixed point of L, meaning L(A) = A. The collection of all Lclosed subsets, denoted LClos(X,L), forms a natural substructure under inclusion.

This framework generalizes the familiar notions of closed and open sets. If L is the standard topological closure operator cl, then Lclosed sets are exactly the closed sets. If, instead, L is the interior operator i, then Lclosed sets are precisely the open sets, since i(A) = A characterizes openness. In other settings, L may be chosen to encode additional structure or constraints, yielding families of sets that reflect those rules. For example, in a graph- or relation-based context, L can express reachability or adjacency constraints, producing lclosed sets that capture those connectivity properties.

Properties of LClos(X,L) depend on the chosen L. In typical cases, the family of Lclosed sets is closed under arbitrary intersections and under unions that are compatible with L’s distributive behavior. Thus, Lclosures offer a flexible, unified language for discussing fixed-point subsets arising from various closure-like operators.

In practice, lcloseds appear in discussions of generalized topology, lattice theory, and formal concept analysis as a way to study fixed-point structures beyond standard closures. See also closure operator, fixed point, open and closed sets, and interior operators.