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Hyperfinitely Generated Algebra

Let X be an infinite set of urelements, and let V(^*X) be an enlargement of V(X). Let H in V(^*X) be an algebra. Then H is hyperfinitely generated provided that it has a hyperfinite subset G such that H is the smallest internal subalgebra of H that contains G. (G is a hyperfinite generating set for H in this case.)

See also

Enlargement, Urelement

This entry contributed by Matt Insall (author's link)

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References

Albeverio, S.; Fenstad, J.; Hoegh-Krohn, R.; and Lindstrøom, T. Nonstandard Methods in Stochastic Analysis and Mathematical Physics. New York: Academic Press, 1986.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.Gonshor, H., "Enlargements Contain Various Kinds of Completions". In Proc. 1972 Victoria Symposium on Nonstandard Analysis. New York: Springer-Verlag, pp. 60-70, 1974.Gonshor, H., "Enlargements of Boolean Algebras and Stone Spaces". Fund. Math. 100, 35-59, 1978.Hurd, A. E. and Loeb, P. A. An Introduction to Nonstandard Real Analysis. Orlando, FL: Academic Press, 1985.Insall, M. "Nonstandard Methods and Finiteness Conditions in Algebra." Zeitschr. f. Math., Logik, und Grundlagen d. Math. 37, 525-532, 1991.Insall, M. "Some Finiteness Conditions in Lattices Using Nonstandard Proof Methods." J. Austral. Math. Soc. 53, 266-280, 1992.Luxemburg, W. A. J. Applications of Model Theory to Algebra, Analysis, and Probability. New York: Holt, Rinehart, and Winston, 1969.Robinson, A. Nonstandard Analysis. Amsterdam, Netherlands: North-Holland, 1966.Schmid, J. "Completing Boolean Algebras by Nonstandard Methods." Zeitschr. für Math. Logik u. Grundlagen der Mathematik 20, 47-48, 1974.Schmid, J. "Nonstandard Constructions for Join-Extensions of Lattices." Houston J. Math. 3, 423-439, 1977.

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Hyperfinitely Generated Algebra

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Insall, Matt. "Hyperfinitely Generated Algebra." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/HyperfinitelyGeneratedAlgebra.html

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