# Monotone function

https://arbital.com/p/poset_monotone_function

by Kevin Clancy Jul 22 2016 updated Dec 3 2016

An order-preserving map between posets.

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];
style = filled;
color = lightgrey;
fontcolor = black;
label = "P";
labelloc = b;
b -> a;
c -> a;

}
subgraph cluster_Q {
node [style=filled];
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];
style = filled;
color = lightgrey;
fontcolor = black;
label = "P";
labelloc = b;
a -> b;
}

subgraph cluster_Q {
node [style=filled];
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 $\phi(a) = v \parallel_Q u = \phi(b)$.