Proposition
167
Suppose that the function \(f:A× A→ B\) is invariant for the equivalence relation \(∼\) in all its variables, i.e.
\[ ∀ x,y,v,w∈ A, \quad x ∼ y∧ v∼ w⇒ f(x,v)=f(y,w)\quad ; \]let \(\tilde f\) be the projection to the quotient \(\widetilde f:\frac A∼× \frac A∼→ B\) that satisfies
\[ f(x,y)=\widetilde f(𝜋(x),𝜋(y))\quad . \]If \(f\) is commutative (resp. associative) then \(\widetilde f\) is commutative (resp. associative).
If \(R\) is a relation in \(A× A\) invariant for \(∼\), and \(R\) is reflexive (resp symmetrical, antisymmetric, transitive) then \(\widetilde R\) is reflexive (resp symmetrical, antisymmetric, transitive).
If \(A\) and \(B\) are ordered and the order is invariant, and \(f\) is monotonic, then \(\widetilde f\) is monotonic.