Independence and duality. (English) Zbl 0782.08001

Both, N.

Math., Rev. Anal. Numér. Théor. Approximation, Math. 33(56), No. 1-2, 9-12 (1991).

Let \(J: P(M)\mapsto P(M)\) be a closure operator on \(M\). A subset \(X\subseteq M\) is called \(J\)-independent if for all \(Y,Z\subseteq X\): \(J(Y)= J(Z)\Rightarrow Y= Z\). In this paper, the author formulates some properties of \(J\)-independence and generalizes them to a mapping (even to an arbitrary binary relation). The relational (functional) properties reveal the duality character of independence and the generating-process.

Reviewer: D.Busneag (Craiova)

MSC:

08A02	Relational systems, laws of composition
06A15	Galois correspondences, closure operators (in relation to ordered sets)
03E20	Other classical set theory (including functions, relations, and set algebra)

Keywords:

function; closure operator; \(J\)-independence; binary relation; duality