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.
4. For algebras 
,
...
,
we have
 |
(3)
|
