Definition 1. A structure is a set together with an -ary operation
We can characterize an operation on a structure: let be an -ary operation on is associative if
where for The statement is trivial in the nullary and unary case. In the binary case we have
or in the ternary case
More generally if is an -ary operation, and is an -ary operation, we say associates with if
Thus an associative operation is one that associates with itself.
Definition 2. A nullary operation that associates with an -ary operation is called an identity with respect to
Definition 3. Let have an identity with respect to an -ary operation We say an -ary operation is unital with respect to if:
Example 4. The unary inverse operation in a group is unital with respect to the binary product.
We say an -ary operation is commutative if
where is a permutation of
Now let be an -ary operation. is left-distributive with respect to if
and right-distributive with respect to if
is distributive with respect to if it is left and right distributive with respect to
Example 5. In a ring, multiplication is distributive with respect to addition ( and ). Addition does not distribute with respect to multiplication though, since that would mean and (Note it does hold in quotient rings: ).
Definition 6. The signature of a structure is the sequence where is the number of -ary operations on Let be treated as an ordinal and be a bijection. We say for an ordinal and -ary operation If we wish to specify what arity of operation we are discussing, we may write instead of to clarify that we mean the th -ary operation on We call the map a -ary operational ordering. A sequence of -ary operational orderings on a structure is called an operational ordering.
If and are structures with the same signature and each have operational orderings, then there is bijective map between each sets’ operations. It simply sends the th -ary operation of to the th -ary operation of To distinguish between operations in the structure and costructure, we write
Specifically if are the operational orderings of and (i.e. and ) we may write as a map
where is the set of -ary operations on and are those on with the map defined by
since if is some arbitrary operation in we have
Definition 7. Let and be structures with the same signature and possess operational orderings and respectively. A map is an –homomorphism, or homomorphism for short, if
for all and all is a strong homomorphism if
where is any -ary operation on and is any -ary operation on An -homomorphism (or strong) is an –isomorphism (or strong isomorphism) if it is bijective.
The latter type of homomorphism implies the former (for any choice of operational orderings). Most homomorphisms studied in algebra are the former type.
Example 8. A ring homomorphism is a homomorphism where and A strong homomorphism between rings would thus additionally require Hence
So is an identity with respect to multiplication on the range. Note that on the range we therefore have
Thus in particular a strong isomorphism of rings implies Such a triviality does not always occur in general with strong isomorphisms since an arbitrary structure may not have any identities. But we do have the following.
Proposition 9. If is a strong isomorphism between structures, then and for any and choice of
Proof. We have
as well as
which implies on the image of But since is bijective, we have for all elements in For the first case we simply have on for all
Thus if is a strong isomorphism, then and have a reduced signature where for all
Proposition 10. Let be an isomorphism.
- If associates with then associates with
- If is commutative, then is commutative.
- If is left (right) distributive with respect to then is left (right) distributive with respect to
- If is an identity with respect to then is an identity with respect to In particular if is unital with respect to then is unital with respect to
These are easy to show and just require a bit of index swapping and parentheses galore, so I leave them to the reader.