Monotonic functions on partially ordered sets (Q1057290)

Theorem: Let (E,\(\leq)\) be an ordered set (in general the ordering is partial) such that for any a,b\(\in E\) there are elements \(m=m(a,b)\), \(M=(a,b)\) in E such that \(m\leq a,b\leq M\); let Y be any totally ordered set. If a function f from E into Y is monotonic on every 3-point chain \(\subset E\), then f is monotonic in (E,\(\leq)\). Five propositions, linked with the Theorem, are proved. Proposition 4: If n is an integer \(>1\), then in the cardinal (coordinate) ordering of \(R^ n\) there is no non- degenerate interval in \(R^ n\) which could be mapped continuously and injectively into R. Pr. 5: If a finitely additive interval real function f in R is locally of constant sign, then f is of constant sign in \(R^ n\).