Internally Extendable Homomorphism

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.