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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.