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 .

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.