Locally Realized Covering Relation
Let 
be a lattice (or a bounded lattice or a complemented lattice, etc.), and let 
be the covering relation of 
:
![C_L=<span style=]() {(x,y) in L^2|x covers y or y covers x}. " src="https://mathworld.wolfram.com/images/equations/LocallyRealizedCoveringRelation/NumberedEquation1.gif" style="height:21px; width:247px" /> |
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.
