Sequential convergences in \(l\)-groups without Urysohn’s axiom. (English) Zbl 0770.06008

The system \(\text{Conv }G\) of all sequential convergences on an \(\ell\)- group \(G\) satisfying Urysohn’s axiom was investigated earlier by the present author [ibid. 39, 717-729 (1989; Zbl 0713.06009)] and others [M. Harminc, ibid. 39, 232-238 (1989; Zbl 0681.06007)]. For an Abelian \(\ell\)-group \(G\) the author studies \(\text{conv } G\), the system of all sequential convergences on \(G\) which do not satisfy Urysohn’s axiom. The \(o\)-convergence on \(G\) belongs to \(\text{conv } G\) but not to \(\text{Conv } G\). There are no atoms in \(\text{conv } G\) unlike the case for \(\text{Conv } G\). The following are equivalent: \(\text{conv } G\) is a lattice, \(\text{conv } G\) is a complete lattice, \(\text{conv } G\) has a greatest element, \(\text{Conv } G\) has a greatest element. If \(G\) is an \(\ell\)-group having finite breadth, then (1) \(\text{conv } G=\text{Conv } G\), and (2) \(G\) is completely distributive. The system \(\text{conv } G\backslash\text{Conv }G\) is also studied.


06F20 Ordered abelian groups, Riesz groups, ordered linear spaces
06F15 Ordered groups
Full Text: DOI EuDML


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