Lattice ordered groups having a largest convergence. (English) Zbl 0713.06009

For an abelian lattice ordered group G let Conv(G) be the system of all sequential convergences on G [cf., M. Harminc, ibid. 37, 533-546 (1987; Zbl 0645.06006)]. Define b(G) to be the set of all bounded sequences in G and let \(Conv_ b(G)=\{\alpha \in b(G):\alpha\in Conv(G)\}\). Let T denote the class of all \(\ell\)-groups G such that Conv(G) possesses a largest element. The present author has shown previously that if the \(\ell\)-group G is archimedean and completely distributive, then \(T\in G\). In the paper under review, the author shows that membership in T is a lattice property, that is, if the \(\ell\)-groups \(G_ 1\) and \(G_ 2\) are isomorphic as lattices (not necessarily as \(\ell\)-groups) and if \(G_ 1\in T\), then \(G_ 2\in T\). The class T is a radical class but not a variety. The partially ordered set \(Conv_ b(G)\) has a largest element if and only if \(G\in T\). If H is an \(\ell\)-group, then the radical T(H) of H corresponding to the radical class T is a closed \(\ell\)-subgroup of H. A notion of homogeneous convergence is defined and examined.
Reviewer: S.P.Hurd


06F20 Ordered abelian groups, Riesz groups, ordered linear spaces


Zbl 0645.06006
Full Text: EuDML


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