Lattice ordered groups having a largest convergence.

*(English)*Zbl 0713.06009For 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 |

##### Keywords:

abelian lattice ordered group; sequential convergences; radical class; homogeneous convergence
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\textit{J. Jakubík}, Czech. Math. J. 39(114), No. 4, 717--729 (1989; Zbl 0713.06009)

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##### References:

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[2] | Harminc M.: The cardinality of the system of all sequential convergences on an abelian lattice ordered group. Czechoslov. Math. J. 37, 1987, 533-546. · Zbl 0645.06006 |

[3] | Harminc M.: Sequential convergences on lattice ordered groups. Czechoslov. Math. J. 39, 1989, 232-238. · Zbl 0681.06007 |

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