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.

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.