Hyper-P Algebra
Let 
be a set of urelements that contains the set 
of natural numbers, and let 
be a superstructure whose individuals are in 
. Let 
be an enlargement of 
, and let 
be an algebra. Let 
be a property of algebras, expressed in the first-order language for the superstructure 
. Then 
is a hyper-
-algebra provided that it satisfies 
in 
.
For example, let 
be the property of "being finite." Then 
is expressible in the first-order language for 
, since 
, and a hyper-
algebra is just a hyperfinite algebra. One useful result involving hyperfinite algebras is the following: An algebra 
is locally finite if and only if it has an hyperfinite extension in 
.
For another example, consider the property of being a simple group. Then a hyper-simple group in 
is just a group 
which has exactly two internal normal subgroups, namely the trivial subgroup and the whole group 
. If an internal group is simple, then it is hyper-simple. It is not known if every hyper-simple group is simple.
For any property 
, the following are equivalent:
1. 
is finite generation-hereditary.
2. The following nonstandard characterization holds for 
: For any set 
of urelements, an algebra 
is a local P-algebra if and only if 
has a hyper-
extension in 
.
REFERENCES:
Gehrke, M.; Kaiser, K.; and Insall, M. "Some Nonstandard Methods Applied to Distributive Lattices." Zeitschrifte für Mathematische Logik und Grundlagen der Mathematik 36, 123-131, 1990.
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.
