EDB — 1Z6

[1Z6]

• Suppose that the function \(f:A× A→ A\) is invariant for the equivalence relation \(∼\) in all its variables, in the sense defined in [(2.194)]↺↻ let \(\tilde f\) be the projection to the quotient

  \[ \widetilde f:\frac A∼× \frac A∼→ \frac A∼\quad . \]

  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).

• Consider the ordered sets \((A,≤_ A)\) and \((B,≤_ B)\), let \(f:A→ B\) be a monotonic function; suppose moreover that \(≤_ A\) is invariant with respect to an equivalence relation \(∼\) on \(A\), e and let \(\widetilde f:{\frac{A}∼ }→ B\) be its projection to the quotient: then \(\widetilde f\) is monotonic.