Cauchy complete l-ideals of a lattice ordered group
by
Štefan Černak
Technical University of Košice, Slovakia

Lattice and Order Theory

Vulikh has defined the notion of convergence of sequences with a regulator in a vector lattice V. A "Convergence regulator" depends on a sequence. This type of convergence was studied by Martinez and Koldunov in l-groups. Luxemburg and Zaanen introduced the notion of a convergence with a fixed regulator in V for all seqences in V.

In the present note the notion of convergence with a fixed regulator u (u-convergence) is examined in an Archimedean l-group G. The main results:

The Dedekind completion D(G) of G is u-Cauchy complete (C-complete) and a u-Cauchy completion C(G) of G is an l-subgroup of D(G).

The system of all C(b)-complete l-ideals of G has a greatest element.

An l-ideal of G generated by C-complete l-ideals of G need not be C-complete.

The system of all C-complete l-ideals of G containing a convergence regulator u has a greatest element.

Date received: March 31, 2005