Set mappings of unrestricted order

Abstract

A set mapping on a set S is a function f mapping S into the powerset of S such that x ∉ f(x) for each x in S. The set map f has order θ if θ is the least cardinal such that |f(x)| < θ for each x in S. A subset H of S is free for f if x ∉ f(y) for all x, y in H. In this paper we use classical results about set mappings of large order to investigate conditions which ensure a large free set for set mappings of unrestricted order.

Intersection properties of functions on cardinals

“…We prove an inversion theorem which allows us to apply existing results about set mappings of order K to get results about set mappings of unrestricted order. See [3].…”