Lattice betweenness relation and a generalization of König’s lemma. (English) Zbl 0890.06001

A tree is a partially ordered set \((T,\leq )\) such that for every \(x\in T\), the set \(\{y\in T: y<x\}\) is well-ordered. Equivalently, a tree is a transitive \(\alpha\)-partite König graph \(G\) for some ordinal \(\alpha \). König’s lemma states that every transitive \(\omega \)-partite König graph \(G\) with finite parts contains an \(\omega\)-frame. An extension of König’s lemma which has the origin in a characterization of lattices by a ternary relation (the lattice betweenness relation) given by M. Kolibiar is presented in the paper. This generalization of König’s lemma states that for every up-directed partially ordered set \(S\), each transitive \(S\)-partite König graph \(G\) with sufficiently many finite parts contains an \(S\)-frame. As an example, this result is applied in lattice theory.


06A06 Partial orders, general
05C20 Directed graphs (digraphs), tournaments
06B05 Structure theory of lattices
08A02 Relational systems, laws of composition
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