Complementing lattice-ordered groups: The finite-valued case (Q1842600)

This article presents necessary and sufficient conditions for a finite- valued \(\ell\)-subgroup to admit a complementation: For a finite-valued \(\ell\)-group \(G\) the following assertions are equivalent: 1. \(G\) admits a simple complementation in which it is convex. 2. \(G\) admits a simple complementation. 3. \(G\) admits a complementation. 4. \(G\) is locally complemented. 5. Every z-subgroup in \(G\) is an intersection of minimal primes. 6. \(G\) is finitely branched and every bounded chain of maximal lex- subgroups is finite. Let us recall that an \(\ell\)-group \(G\) is complemented if for each \(a\in G\) there exists \(b\in G\) such that \(| a| \wedge | b| =0\), while \(| a|\vee | b|\) is a unit of \(G\). If \(G\) is an \(\ell\)-subgroup of a complemented \(\ell\)-group \(H\), then \(H\) is a complementation of \(G\) if the convex hull \(G^ C\) is a strongly rigid extension of \(G\) and a z-subgroup of \(H\) (i.e., \(x\in G^ C \Rightarrow x^{\perp\perp} \subseteq G^ C\)). An \(\ell\)-group \(G\) is finite-valued if every non-zero element \(a\in G\) has at most a finite number of values (a maximal convex \(\ell\)-subgroup in \(G\) with respect to excluding the element \(a\)).