# Maximal and minimal elements, definition

Let \( X \) be a partially ordered set, and let \( Y \) be a subset of \( X \).

#### Minimal element

We say that \( y \) is a minimal element of \( Y \) if \( y \in Y \) and there is no element \( y' \in Y \) such that \( y' \le y \).

#### Maximal element

We say that \( y \) is a maximal element of \( Y \) if \( y \in Y \) and there is no element \( y' \in Y \) such that \( y \le y' \).

The definition cannot be: for all \(y \in Y\) \(y' \le y\), as this would force there to be a total order (fully connected graph).

#### Maximal element vs supremum

A *maximal element* can be thought of as
an endmost element in a directed acyclic graph. The relation that defines this
graph can be anything that meets the criteria needed to produce partial or total
ordering. The maximal element must exist *in the set*. A set
may have zero, one or multiple maximal elements (image multiple connected
components of a directed graph).

*Supremum* is a concept defined only for sets of reals using the
predicate \( x \le y \iff ((x - y) \text{ is nonpositive}) \). The supremum of a set
of reals *does not need to be within the set*. A set can only
have one supremum, and it always has one supremum (\( \infty \) is used for
sets when there is no least upper bound).

### Example

\( X = \{ \{1, 2\}, \{ 2 \}, \{2, 3\}, \{2, 3, 4\}, \{ 5 \} \} \) and the set inclusion relation \( \subseteq \) has the following minimal and maximal elements:

- minimal
- \( \{2\}, \{5\} \)
- maximal
- \( \{1, 2 \}, \{ 2, 3, 4\} \)
- both
- \( \{5\} \)
- neither
- \( \{2, 3\} \)