Ordering the set of antichains of an ordered set
type: Item
instance of: scholarly article
title: Ordering the set of antichains of an ordered set
zbMATH Open document ID: 0840.06002
published in: Collectanea Mathematica
publication date: 1996-07-01T00:00:00Z
full work available at URL: https://eudml.org/doc/39873
review text: Let \((S, \leq)\) be an ordered set, \(A(S)\) the set of antichains on \(S\). If \(X,Y \in A (S)\), let \(X \leq Y\) when for every \(x \in X\) there exists \(y \in Y\) such that \(x \leq y\). Then \(\leq\) is an order relation. Let \(F(S)\) be the set of finite antichains on \(S\). Then \((A(S), \leq) \) is a sup-lattice with \(\sup \{X,Y\} = \text{Max} \{X \cup Y\}\) and \((F(S), \leq)\) is a sub-sup-lattice, enjoying a universal property. If \((S, \leq)\) is noetherian with smallest element, then \((A (S), \leq)\) is a complete lattice, enjoying a universal property. It is also shown that if \((S, \leq)\) is artenian and narrow, then so is \((F(S), \leq)\). In \S 2, \((S,+, \leq)\) is an additive commutative ordered monoid. If \(X,Y \in F(S)\), let \(X {\underset {a} +} Y = \text{Max} (X + Y)\). Then if \((S,+, \leq)\) is strictly ordered, then \((F(S),+, \leq)\) is an ordered monoid and several properties are established. In \S 3, let \((S,\leq)\) be an ordered set, let \((R,+)\) be an abelian group. If \(X, Y \in F(S)\) let \(X \triangleleft Y\) when for every \(x \in X\) there exists \(y \in Y\) such that \(y \leq x\). Let \(A\) be the set of all \(f : S \to R\) such that \(\text{supp} (f) = \{s \in S \mid f(s) \neq 0\}\) is artinian and narrow. Then \(A\) is an abelian additive group. Let \(d : A \times A \to (F(S) \cup \{ \emptyset\}, \triangleleft)\) be defined by \(d(f,g) = \text{Min}_\leq (\text{supp} (f-g))\). Then \(d\) is an ultrametric distance, compatible with the addition. If \((S,+, \leq)\) is a strictly ordered monoid, then \(A\) is a ring, with multiplication \(f*g\) given by \((f*g) (s) = \sum_{t + u = s} f(t) g(u) \). Then \(d\) satisfies \(d(f*h, g*h) \leq d(f,g){\underset {a} +}|h |\) where \(|h |= d(h,0)\). Various properties of the ultrametric distance are indicated.
Mathematics Subject Classification ID: 06A06
Mathematics Subject Classification ID: 06F05
zbMATH DE Number: 869953
zbMATH Keywords: ordered set
zbMATH Keywords: antichains
zbMATH Keywords: ordered monoid
zbMATH Keywords: ultrametric distance
author: Paulo Ribenboim
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 869953