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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

##### MSC:
 06F20 Ordered abelian groups, Riesz groups, ordered linear spaces
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##### References:
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