Monotone function: exercises

Try these exercises and become a deity of monotonicity.

Monotone composition

Let \(P, Q\), and \(R\) be posets and let \(f : P \to Q\) and \(g : Q \to R\) be monotone functions. Prove that their composition \(g \circ f\) is a monotone function from \(P\) to \(R\).

Evil twin

Let \(P\) and \(Q\) be posets. A function \(f : P \to Q\) is called antitone if it reverses order: that is, \(f\) is antitone whenever \(p_1 \leq_P p_2\) implies \(f(p_1) \geq_Q f(p_2)\). Prove that the composition of two antitone functions is monotone.

Partial monotonicity

A two argument function \(f : P \times A \to Q\) is called partially monotone in the 1st argument whenever \(P\) and \(Q\) are posets and for all \(a \in A\), \(p_1 \leq_P p_2\) implies \(f(a, p_1) \leq_Q f(a, p_2)\). Likewise a 2-argument function \(f : A \times P \to Q\) is called partially monotone in the second argument whenever \(P\) and \(Q\) are posets and for all \(a \in A\), \(p_1 \leq_P p_2\) implies \(f(p_1, a) \leq_Q f(p_2, a)\).

Let \(P, Q, R\), and \(S\) be posets, and let \(f : P \times Q \to R\) be a function that is partially monotone in both of its arguments. Furthermore, let \(g_1 : S \to P\) and \(g_2 : S \to Q\) be monotone functions.

Prove that the function \(h : S \to R\) defined as \(h(s) \doteq f(g_1(s), g_2(s))\) is monotone.

Brain storm

List all of the commonly used two argument functions you can think of that are partially monotone in both arguments. Also, list all of the commonly used two argument functions you can think of that are partially monotone in one argument and partially antitone in the other.