Reflexive Index of a Family of Sets

Abstract

As a further study on re∞exive families of subsets, we introduce the re∞exive index for a family of subsets of a given set and show that the index of a flnite family of subsets of a flnite or countably inflnite set is always flnite. The re∞exive indices of some special families are also considered. Given a set X, let Sub(X) denote the set of all subsets of X and End(X) denote the set of all endomappings f : X i! X. For any A µ Sub(X) and F µ End(X) deflne Alg(A) = ff 2 End(X) : f(A) µ A for all A 2 Ag; Lat(F) = fA 2 Sub(X) : f(A) µ A for all f 2 Fg: A family A µ Sub(X) is called re∞exive if A = Lat(Alg(A)), or equivalently, A = Lat(F) for some F µ End(X). As was shown in (9), A µ Sub(X) is re∞exive ifi it is closed under arbitrary unions and intersections and contains the empty set and X. The re∞exive families F µ End(X) were also introduced and characterized as those subsemigroups L of (End(X);-) such that L is a lower set and contains all existing suprema of subsets of L with respect to a naturally deflned partial order on End(X). The similar work in functional analysis is on the re∞exive invariant subspace lattices and re∞exive operator algebras (1-6). For any A µ Sub(X), let ^ A = Lat(Alg(A)). Then ^ A is the smallest family of subsets containing A which is closed under arbitrary unions and intersections containing empty set ; and X, and ^