Ordering the set of antichains of an ordered set
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.
scientific article; zbMATH DE number 869953