Internally Extendable Homomorphism

Let X be an infinite set of urelements, and let V(^*X) be an enlargement of the superstructure V(X). Let A,B in V(X) be finitary algebras with finitely many operations, and let A^^ and B^^ be their extension monads in V(^*X). Let h:A^^->B^^ be a homomorphism. Then h is internally extendable provided that there is an internal subalgebra A_1 of ^*A which contains A^^ and there is a homomorphism gamma:A_1->^*B such that if a in A^^, then gamma(a)=h(a).

For a homomorphism h:A^^->B^^, the following are equivalent:

1. h is internally extendable and h[A] is a subalgebra of B,

2. For some homomorphism g:A->B, h=g^^ is the restriction to A^^ of ^*g.

This entry contributed by Matt Insall (author's link)

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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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Internally Extendable Homomorphism

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Insall, Matt. "Internally Extendable Homomorphism." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein.

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