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# Saturated Enlargement

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

For each internal binary relation , and each set , if is contained in the domain of and the cardinality of is less than , then there exists a in the range of such that if , then .

If is -saturated for some cardinal that is greater than or equal to the cardinality of , then we just say that is saturated. If it is -saturated for some cardinal that is greater than or equal to the cardinality of , then we say it is polysaturated.

Let be the set of real numbers, as urelements. Let be a cardinal number that is larger than the cardinality of the power set of , and let be a -saturated enlargement of . Let be an internal subset of , and let . Then is a closed subset of (in the usual topology of the real numbers).

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

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

Enlargement

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

## References

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.

## Referenced on Wolfram|Alpha

Saturated Enlargement

## Cite this as:

Insall, Matt. "Saturated Enlargement." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/SaturatedEnlargement.html