Monotone function: exercises

Try these exercise to 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 $f\circ g$ is a monotone function from $P$ to $Q$.

Antitone composition

Let $⟨P,{\le }_P⟩,⟨Q,{\le }_Q⟩$ be posets. A function $f:P\to Q$ is called antitone if it reverses order: that is, $f$ is antitone whenever $p_1{\le }_P{p}_2$ implies $f\left(p_1\right){\ge }_{Q}f\left(p_2\right)$. Prove that the composition of two antitone functions is monotone.

Partial monotonicity

An 2-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 $a\in A$, $p_1{\le }_P{p}_2$ implies $f(a,p_1){\le }_{Q}f(a,p_2)$. 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 $a\in A$, $p_1{\le }_P{p}_2$ implies $f(p_1,a){\le }_{Q}f(p_2,a)$.

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 $h\left(s\right)\doteq f\left(g_1\right(s),g_2(s\left)\right)$ is monotone.