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 .
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.
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 
.
, the following are equivalent:
is internally extendable and ![h[A]](/images/equations/InternallyExtendableHomomorphism/Inline18.svg)
is a subalgebra of 
,
, 
is the restriction to 
of 
.
