Some Finiteness Conditions in Lattices Using Nonstandard Proof Methods
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.
M. Insall, "Some Finiteness Conditions in Lattices Using Nonstandard Proof Methods," Journal of the Australian Mathematical Society, Series A, American Mathematical Society, Jan 1992.
Mathematics and Statistics
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© 1992 American Mathematical Society, All rights reserved.