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Let be an infinite set of urelements, and let be an enlargement of the superstructure . Let be finitary algebras with finitely many operations, and let and be their extension monads in . Let be a homomorphism. Then is internally extendable provided that there is an internal subalgebra of which contains and there is a homomorphism such that if , then .
For a homomorphism , the following are equivalent:
1. is internally extendable and is a subalgebra of ,
2. For some homomorphism , is the restriction to of .
REFERENCES:
Albeverio, S.; Fenstad, J.; Hoegh-Krohn, R.; and Lindstrøom, T. Nonstandard Methods in Stochastic Analysis and Mathematical Physics. New York: Academic Press, 1986.
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.
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