Let and $⟨ Q, \leq_{Q} ⟩$ be posets. Then a function $ϕ : P \to Q$ is said to be monotone (alternatively, order-preserving) if for all $s, t \in P$ , $s \leq_{P} t$ implies $ϕ (s) \leq_{Q} ϕ (t)$ .

Positive example

A simple monotone map phi

dot source:

digraph G { node = 0.1, height = 0.1 rankdir = BT; rank = same; compound = true; fontname="MathJax_Main";

subgraph cluster_P { node [style=filled,color=white]; edge = "none"; style = filled; color = lightgrey; fontcolor = black; label = "P"; labelloc = b; b -> a; c -> a;

} subgraph cluster_Q { node [style=filled]; edge = "none"; color = black; fontcolor = black; label= "Q"; labelloc = b; u -> t; } edge = blue, style = dashed fontcolor = blue; label = "φ";
labelloc = t; b -> t = false; a -> t = false; c -> u = false; }

Here is an example of a monotone map $ϕ$ from a poset $P$ to another poset $Q$ . Since $\leq_{P}$ has two comparable pairs of elements, $(c, a)$ and $(b, a)$ , there are two constraints that $ϕ$ must satisfy to be considered monotone. Since $c \leq_{P} a$ , we need $ϕ (c) = u \leq_{Q} t = ϕ (a)$ . This is, in fact, the case. Also, since $b \leq_{P} a$ , we need $ϕ (b) = t \leq_{Q} t = ϕ (a)$ . This is also true.

Negative example

A simple, non-monotone map

dot source:

digraph G { node = 0.1, height = 0.1 rankdir = BT; rank = same; compound = true; fontname="MathJax_Main";

subgraph cluster_P { node [style=filled,color=white]; edge = "none"; style = filled; color = lightgrey; fontcolor = black; label = "P"; labelloc = b; a -> b; }

subgraph cluster_Q { node [style=filled]; edge = "none"; color = black; fontcolor = black; label= "Q"; labelloc = b; w -> u; w -> v; u -> t; v -> t; } edge = blue, style = dashed fontcolor = blue; label = "φ";
labelloc = t; b -> u = false; a -> v = false; }

Here is an example of another map $ϕ$ between two other posets $P$ and $Q$ . This map is not monotone, because $a \leq_{P} b$ while $ϕ (a) = v ∥_{Q} u = ϕ (b)$ .

Additional material

For some examples of montone functions and their applications, see Monotone function: examples. To test your knowledge of monotone functions, head on over to Monotone function: exercises.