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 .