On Riesz groups
The aim of this paper is to prove the following Theorem 1 and to show some of its consequences and some of its connections with the current literature. Theorem 1. Let \(G\) be a directed po-group and let \(H\) be an o-ideal of \(G\) such that every member of \(H^ +\) is primal in \(G\). If \(G/H\) is a Riesz group, then so is \(G\). (\(x\in G\) is primal in \(G\) if \(x\in G^ +\) and \(x\leq a_ 1+ a_ 2\), where \(a_ i\in G^ +\), implies \(x= x_ 1+ x_ 2\) such that \(0\leq x_ i\leq a_ i\).) This result is a general group-theoretic version of Cohn's analogue of Nagata's UFD's Theorem for Schreier domains. The other point of interest is the following Corollary: Let \(G\) be a po-Schreier extension of a directed po-group \(H\) by another po-group \(F\). Then \(G\) is a Riesz group iff the following hold: (1) Every member of \(H^ +\) is primal in \(G\); (2) \(F\) is a Riesz group. The main result (Section 1). Theorem 7. Every directed po-group \(G\) contains an interpolating o-ideal \(M=M(G)\) that is the largest (that is if \(M_ 1\) is an interpolating o- ideal of \(G\) then \(M_ 1\subseteq M(G)\)). Consequently \(G/M\) contains no non-trivial completely primal elements. (An o-ideal \(H\) of \(G\) is an interpolating o-ideal of \(G\) if every element of \(H^ +\) is a primal element of \(G\). \(x\in G^ +\) is called completely primal if every element in \([0,x]\) is primal.) The Sections 2 and 3 are devoted to indicating the connection of Theorem 1 with Cohn's results.
scientific article; zbMATH DE number 492781