Saturated Enlargement

Let X be a set of urelements, and let V(X) be the superstructure with X as its set of individuals. Let kappa be a cardinal number. An enlargement V(^*X) is kappa-saturated provided that it satisfies the following:

For each internal binary relation r in V(^*X), and each set A in V(^*X), if A is contained in the domain of r and the cardinality of A is less than kappa, then there exists a y in the range of r such that if x in A, then (x,y) in r.

If V(^*X) is kappa-saturated for some cardinal kappa that is greater than or equal to the cardinality of ^*X, then we just say that V(^*X) is saturated. If it is kappa-saturated for some cardinal kappa that is greater than or equal to the cardinality of V(X), then we say it is polysaturated.

Let R be the set of real numbers, as urelements. Let kappa be a cardinal number that is larger than the cardinality of the power set of R, and let V(^*R) be a kappa-saturated enlargement of V(R). Let B be an internal subset of ^*R, and let st(B)={x in R| for some y in B,x=st(y)}. Then st(B) is a closed subset of R (in the usual topology of the real numbers).

Using saturated enlargements, one may prove the following result in universal algebra:

Let V be a variety that satisfies the property that for each subvariety W of V, and each algebra A in V, if A is generated by its W-subalgebras, then A in W. Then any V-sum of locally finite algebras is locally finite.

See also


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

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Albeverio, S.; Fenstad, J.; Hoegh-Krohn, R.; and Lindstrøom, T. Nonstandard Methods in Stochastic Analysis and Mathematical Physics. New York: Academic Press, 1986.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.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.

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Saturated Enlargement

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Insall, Matt. "Saturated Enlargement." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein.

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