Some Finiteness Conditions in Lattices Using Nonstandard Proof Methods

Citation data:

Journal of the Australian Mathematical Society

Publication Year:
1992

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Repository URL:
http://scholarsmine.mst.edu/math_stat_facwork/606
Author(s):
Insall, Matt
Publisher(s):
American Mathematical Society
Tags:
Mathematics; Statistics and Probability
article description
One of the early themes in nonstandard analysis is a characterization of hereditary finite properties of algebraic structures in terms of their hyperfinite extensions. The results of this type, practically always obtained as a simple consequence of upward and downward transfer principles, are used here in the category of lattices for analyzing properties such as existence of (local) polarities. A typical result (Proposition 2.1) says that a pair of functions f, g:L ! L is a local polarity of L if and only if L has a hyperfinitely generated extension L_ for which (_f|L_, _g|L_) is a (hyper)polarity of L_. Among other properties of lattices analyzed from this point of view are tightness, 0, 1-simplicity, etc.