Monotonic functions on partially ordered sets
type: Item
instance of: scholarly article
title: Monotonic functions on partially ordered sets
zbMATH Open document ID: 0563.06002
DOI: 10.1016/0097-3165(84)90049-9
author: H. Burkill
published in: Journal of Combinatorial Theory. Series A
publication date: 1984-01-01T00:00:00Z
review text: 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 \(R^ n\).
Mathematics Subject Classification ID: 06A06
Mathematics Subject Classification ID: 03E20
zbMATH DE Number: 3896986
zbMATH Keywords: ordered set
zbMATH Keywords: totally ordered set
zbMATH Keywords: finitely additive interval real function
MaRDI profile type: Publication
full work available at URL: 0097-3165(84)90049-9
OpenAlex ID: W1964855007
cites work: SOME REMARKS ON INTERVAL FUNCTIONS
cites work: Partitions of sets of matrices
description: scientific article; zbMATH DE number 3896986