Unique homomorphic extension theorem

The unique homomorphic extension theorem is a result in mathematical logic which formalizes the intuition that the truth or falsity of a statement can be deduced from the truth values of its parts.

The lemma

Let A be a non-empty set, X a subset of A, F a set of functions in A, and the inductive closure of X under F.
Let be B any non-empty set and let G be the set of functions on B, such that there is a function in G that maps with each function f of arity n in F the following function in G.
From this lemma we can now build the concept of unique homomorphic extension.

The theorem

If is a free set generated by X and F, for each function there is a single function such that:
For each function f of arity n > 0, for each

Consequence

The identities seen in e show that is an homomorphism, specifically named the unique homomorphic extension of. To prove the theorem, two requirements must be met: to prove that the extension exists and is unique.

Proof of the theorem

We must define a sequence of functions inductively, satisfying conditions and restricted to. For this, we define, and given then shall have the following graph:
First we must be certain the graph actually has functionality, since is a free set, from the lemma we have when, so we only have to determine the functionality for the left side of the union. Knowing that the elements of G are functions, the only instance where and for some is possible is if we have for some and for some generators and in.
Since and are disjoint when this implies and. Being all in, we must have.
Then we have with, displaying functionality.
Before moving further we must make use of a new lemma that determines the rules for partial functions, it may be written as:
Be a sequence of partial functions such that. Then, is a partial function.
Using, is a partial function. Since then is total in.
Furthermore, it is clear from the definition of that satisfies and. To prove the uniqueness of, or any other function that satisfies and, it is enough to use a simple induction that shows and work for , and such is proved the Theorem of the Unique Homomorphic Extension.

Example of a particular case

We can use the theorem of unique homomorphic extension for calculating numeric expressions over whole numbers. First, we must define the following:
Be
Be he inductive closure of under and be
Be
Then will be a function that calculates recursively the truth-value of a proposition, and in a way, will be an extension of the function that associates a truth-value to each atomic proposition, such that: