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

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dot source:

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    c -> a;

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  label = "φ";  
  labelloc = t; 
  b -> t [constraint = false];
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  c -> u [constraint = false];
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%%

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.