## Independence and duality.(English)Zbl 0782.08001

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.

### 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)