Maximal and minimal elements.

In this subsection, we consider a poset $(E, \prec )$.

Definition 4.4.5       

1. An element $M$ in $E$ is called a maximal element in $E$ if there exist no $x \in E$ such that $M \prec x$.
2. An element $m$ in $E$ is called a minimal element in $E$ if there exist no $y \in E$ such that $y \prec m$.

The Hasse diagram of a (finite) poset is a useful tool for finding maximal and minimal elements: they are respectively top and bottom elements of the diagram. For example, in $\mathcal{D}_{24}$, $1$ is a minimal element and $24$ is a maximal element. Note, however, that this example is quite special: there is a unique maximal element and a unique minimal element. This situation is not general. Consider the set $E=\{2,3,5,6,8,10,12,15,24 \}$ ordered by division. Its Hasse diagram is displayed in Figure 9.

Figure 9: Hasse diagram of a poset.

It appears clearly that $(E, \; \vert \; )$ has three minimal elements (namely $2$, $3$ and $5$) and three maximal elements (namely $10$, $15$ and $24$).

Definition 4.4.6       

1. An element $a$ is the greatest element of $E$ if, for any element $x \in E$, we have $x \prec a$.
2. An element $b$ is the least element of $E$ if, for any element $y \in E$, we have $b \prec y$.

Example 4.4.7       

1. The set $\mathcal{D}_{24}$ has a least element (namely $1$) and a greatest element (namely $24$).
2. The poset $\{2,3,5,6,8,10,12,15,24 \}$, ordered by division, has neither a least nor a greatest element.
3. The poset $\{ 1,2,3,5,6,8,10,12,15,24 \}$, ordered by division, has $1$ as its least element and has no greatest element.

Definition 4.4.8   Let $(E, \prec )$ be a poset. The set $E$ is well-ordered is every non empty subset of $E$ has a least element.

For example, the set of all natural numbers, ordered by $\leq$ is well-ordered.