Locally Realized Covering Relation

Let  be a lattice (or a bounded lattice or a complemented lattice, etc.), and let  be the covering relation of :

Then  is locally realized provided that for every finite subset  of , there is a finitely generated sublattice  of , that contains , for which . It can be shown that  is locally realized if and only if there is a hyperfinitely generated sublattice  of  such that . Using this characterization of locally realized covering relations, the following standard result can be proved using nonstandard methods:

Let  be a locally finite lattice in which the covering relation is locally realized, and let  be the sublattice of  which is generated by . Then  is a connected tolerance of , and it is in fact the smallest locally subconnected (and locally connected) tolerance of .

REFERENCES:

Burris, S. and Sankappanavar, H. P. A Course in Universal Algebra. New York: Springer-Verlag, 1981. http://www.thoralf.uwaterloo.ca/htdocs/ualg.html.

Gehrke, M.; Kaiser, K.; and Insall, M. "Some Nonstandard Methods Applied to Distributive Lattices." Zeitschrifte für Mathematische Logik und Grundlagen der Mathematik 36, 123-131, 1990.

Grätzer, G. Lattice Theory: First Concepts and Distributive Lattices. San Francisco, CA: W. H. Freeman, 1971.

Grätzer, G. Universal Algebra, 2nd ed. New York: Springer-Verlag, 1979.

Insall, M. "Some Finiteness Conditions in Lattices Using Nonstandard Proof Methods." J. Austral. Math. Soc. 53, 266-280, 1992.