# 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

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;
}

<div>

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

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;
}
<div>

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.

Children:

Parents:

- Order theory
The study of binary relations that are reflexive, transitive, and antisymmetic.