MATH.9:4.1 - State the input, permitted answers and transformation law

Let D be the inputs and Y the possible outputs. Write A(d) for the subset of outputs permitted for input d. An admissible root, minimizing allocation or selected vertex can define this subset.

Give a group G acting on both D and Y. Its action satisfies e*d=d and (g*h)*d=g*(h*d), with analogous rules for outputs; every transformation has an inverse. Different actions on input and output can use the same group elements.

Establish that the permitted answers transform with the input:

A(g*d) = g*A(d),

where g*A(d) means the set of g*y for y in A(d). MATH.8 supplies the solution-preservation argument when it has to be constructed.

The requested deterministic rule f must satisfy both f(d) in A(d) and

f(g*d)=g*f(d).

This relation is equivariance. If the requested output stays unchanged under the transformations, use the trivial output action g*y=y; the relation then expresses invariance.

Keep attributes that the rule is allowed to use inside d. A distinguished label, orientation or mark can change the stabilizer and the existence of a choice. A coordinate label that merely changes under relabeling cannot silently serve as a fixed priority.