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
- \( \{2, 3\} \)