Extension Monad

Let X be a set of urelements, and let V(^*X) be an enlargement of the superstructure V(X). Let A in V(X) be a finitary algebra with finitely many fundamental operations. Then the extension monad (in V(^*X)) of A is the (generally external) subalgebra of ^*A that is given by

 A^^= intersection {B<=^*A:B is an internal extension of A}.

It can be shown that for any such algebra A, we have

 A^^= intersection {B<=^*A:B is an internal hyperfinitely 
 generated extension of A}  
= union {H<=^*A:H is internally generated by 
 a finite subset of A}

and several other interesting characterizations hold for extension monads.

Here are some results involving extension monads:

1. An algebra A is locally finite if and only if A^^=A.

2. For any algebra, the following are equivalent: A is finitely generated, A^^=^*A, and A^^ is internal.

3. Let A and B be algebras, with phi a function from A to B. Then phi is a homomorphism if and only if the restriction phi^^ of ^*phi to A^^ is a homomorphism.

4. For algebras A_1, ...A_n, we have


See also

Superstructure, Urelement

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.Gehrke, M.; Kaiser, K.; and Insall, M. "Some Nonstandard Methods Applied to Distributive Lattices." Zeitschrifte für Mathematische Logik und Grundlagen der Mathematik 36, 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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Extension Monad

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Insall, Matt. "Extension Monad." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein.

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