Let be a set of urelements, and let be an enlargement of the superstructure . Let be a finitary algebra with finitely many fundamental operations. Then the extension monad (in ) of is the (generally external) subalgebra of that is given by
(1)
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It can be shown that for any such algebra , we have
(2)
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and several other interesting characterizations hold for extension monads.
Here are some results involving extension monads:
1. An algebra is locally finite if and only if .
2. For any algebra, the following are equivalent: is finitely generated, , and is internal.
3. Let and be algebras, with a function from to . Then is a homomorphism if and only if the restriction of to is a homomorphism.
4. For algebras , ..., we have
(3)
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