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)

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

(2)

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.

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