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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.

##### MSC:
 06F20 Ordered abelian groups, Riesz groups, ordered linear spaces 06F15 Ordered groups
##### Keywords:
lattice-ordered group; sequential convergence; atoms
##### Citations:
Zbl 0713.06009; Zbl 0681.06007
Full Text:
##### References:
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