Feebly projectable \(\ell \)-groups
The authors remark that, when preparing the present paper, they were motivated by their previous results on frames. Several notions introduced here for \(\ell\)-groups have their counterparts in the theory of frames. An \(\ell\)-group \(G\) is defined to be feebly projectable if whenever \(a,b\in G^+\) are disjoint, then there is \(x\in G^+\) with \(a\in x^{\perp\perp}\), \(b\in x^\perp\), \(G=x^{\perp\perp}\oplus x^\perp\). Further, \(G\) is said to be flatly projectable if for any pair of disjoint elements \(a\) and \(b\), there is a polar subgroup of \(S\) of \(G\) such that \(a\in S\), \(b\in S^\perp\) and \(G=S\oplus S^\perp\). If every prime subgroup of \(G\) contains a minimal prime subgroup then \(G\) is said to have stranded primes. These three notions serve as a main tool in the investigations performed in the paper. Typical results: Suppose that \(\{ G_i\}_{i\in I}\) is a collection of \(\ell\)-groups and \(G=\prod_{i\in I} G_{i}\); then \(G\) is feebly (flatly) projectable if and only if each \(G_i\) is feebly (flatly) projectable. Suppose \(G\) is flatly projectable, then it is representable. Further, the authors prove some related results for Archimedean \(\ell\)-groups with weak order unit and for commutative semiprime \(f\)-rings.
scientific article; zbMATH DE number 5721489
scientific article