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Ball, Richard N.

Distributive Cauchy lattices. (English) Zbl 0539.06008

Algebra Univers. 18, 134-174 (1984).
The author has pioneered the study of completions of lattice ordered groups by means of Cauchy structures. This paper extends this study to arbitrary distributive lattices by means of Cauchy structures obtained from two intrinsic (and generally non-topological) lattice convergences called \(\alpha\) and \(\beta\). The \(\beta\)-convergence, which for infinitely distributive lattices coincides with Birkhoff’s order convergence, leads to a Cauchy completion closely related to (but not always equal to) the MacNeille lattice completion. In the case of \(\alpha\)-convergence, repeated iteration of the Cauchy completion process may be required, but the end result is an ”essential” extension which is an infinitely distributive, complete lattice.
Reviewer: D.C.Kent

Cited in 14 Documents

MSC:

06B30 Topological lattices
06B23 Complete lattices, completions
54A20 Convergence in general topology (sequences, filters, limits, convergence spaces, nets, etc.)
06D10 Complete distributivity
54E52 Baire category, Baire spaces

Keywords:

completions; Cauchy structures; distributive lattices; lattice convergences; order convergence; MacNeille lattice completion
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References:

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