Minimal Element need not be Unique/Examples/Arbitrary Example 1

Example of Minimal Element need not be Unique

Consider the set $S = \set {a, b, c, d, e}$ with the partial ordering $\preccurlyeq$ defined as:

${\preccurlyeq} := \set {\tuple {c, a}, \tuple {d, a}, \tuple {e, a}, \tuple {d, b}, \tuple {e, b}, \tuple {c, b}, \tuple {c, e} }$

This can be illustrated using the following Hasse diagram:

It can be seen by inspection that both $c$ and $d$ are minimal elements of the partially ordered set $\struct {S, \preccurlyeq}$.

Sources

• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): maximal
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): maximal