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Kernels of lattice ordered groups defined by properties of sequences. (English) Zbl 0556.06007
Let \({\mathcal P}\) be a class of lattice ordered groups (\(\ell\)-groups). If each \(\ell\)-group G admits a largest convex \(\ell\)-subgroup C that belongs to \({\mathcal P}\) then we say that C is the \({\mathcal P}\)-kernel of G and the \({\mathcal P}\) has the kernel property. Two kernels of abelian \(\ell\)-groups that are defined by properties of sequences are studied in this paper.
A subset X of an abelian \(\ell\)-group is called Cauchy b-complete if each fundamental sequence in X which is bounded in X order converges to an element of X. The class of Cauchy b-complete \(\ell\)-groups has the kernel property, but the kernel need not be Cauchy complete. For an abelian \(\ell\)-group G let H be the set of all \(v\in G\) such that the interval [0,\(| v|]\) is a Cauchy complete subset of G. Then H is a convex \(\ell\)-subgroup of G.
A subset X of G is L-complete if each fundamental sequence in X laterally converges to an element in X. Let G be a completely representable \(\ell\)- group and let H be the subset of all elements \(a\in G\) such that the interval [0,\(| a|]\) is a complete L-subset of G. Then H is a convex \(\ell\)-subgroup of G.
Reviewer: P.F.Conrad

MSC:
06F15 Ordered groups
06F30 Ordered topological structures
54A20 Convergence in general topology (sequences, filters, limits, convergence spaces, nets, etc.)
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