MATH.2:4.3 - Test compatibility, including the domain of a partial operation

For a total operation f, compare tuples whose corresponding entries are equivalent. The compatibility condition is:

a1~b1, …, an~bn ⇒ f(a1,…,an)~f(b1,…,bn).

An equivalence relation satisfying this condition for every retained total operation is a congruence for those operations.

For a partial operation, first require agreement about whether it can be applied:

a1~b1, …, an~bn ⇒ ((a1,…,an)∈D_f ⇔ (b1,…,bn)∈D_f).

Where both tuples are in the domain, require equivalent outputs as above. This pattern uses the resulting quotient convention in which availability is independent of the representative. A set-valued or approximate account that deliberately combines differing possibilities is another construction.

Use an algebraic argument for a general claim, or exhaustive checking when the specified carrier is finite. A few successful examples can suggest the relation; one failure is enough to refute compatibility.

If the test fails, keep the distinguishing information in the objects or narrow the operation/question being claimed. For a finite partition, split the failing class using the exposed distinction and retest the affected operations. This is a local repair; repeated splitting is not being offered here as a general minimization algorithm.