Preserving meets in meet-dense poset completions

Details Preserving meets in meet-dense poset completions Preserving meets in meet-dense poset completions

2012-01-16

arxiv.org/abs/1201.3285v2

Mathematics

Rings and Algebras

Scientific paper

Defining P* to be the complete lattice of upsets (ordered by reverse inclusion) of a poset P we give necessary and sufficient conditions on a subset S of P* for P to admit a meet-completion e from P to Q where e preserves the infimum of an upwardly closed set from P if and only if it is in S. We show that given S satisfying these conditions the set M of these completions forms a topped weakly lower semimodular lattice. In particular, when P is finite M is a lower semimodular lattice, and a lower bounded homomorphic image of a free lattice. We provide an example where M does not have a bottom element.

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