Monotone function

Let $\langle P, \leq_P \rangle$ and $\langle Q, \leq_Q \rangle$ be posets. Then a function $\phi : P \rightarrow Q$ is said to be monotone (alternatively, order-preserving) if for all $s, t \in P$, $s \le_P t$ implies $\phi(s) \le_Q \phi(t)$.

Positive example

%%comment:
dot source:

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

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

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

%%

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

Negative example

%%comment:
dot source:

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

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

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

Here is an example of another map $\phi$ between two other posets $P$ and $Q$. This map is not monotone, because $a \leq_P b$ while $\phi(a) = v \parallel_Q u = \phi(b)$.

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.