Maximal Element in Toset is Unique and Greatest

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Theorem

Let $\struct {S, \preceq}$ be a totally ordered set.

Let $M$ be a maximal element of $\struct {S, \preceq}$.


Then:

$(1): \quad M$ is the greatest element of $\struct {S, \preceq}$.
$(2): \quad M$ is the only maximal element of $\struct {S, \preceq}$.


Proof

By definition of maximal element:

$\forall y \in S: M \preceq y \implies M = y$

As $\struct {S, \preceq}$ is a totally ordered set, by definition $\preceq$ is a connected.

That is:

$\forall x, y \in S: y \preceq x \lor x \preceq y$

It follows that:

$\forall y \in S: M = y \lor y \preceq M$

But as $M = y \implies y \preceq M$ by definition of $\preceq$, it follows that:

$\forall y \in S: y \preceq M$

which is precisely the definition of greatest element.

Hence $(1)$ holds.

$\Box$


Suppose $M_1$ and $M_2$ are both maximal elements of $S$.

By $(1)$ it follows that both are greatest elements.

It follows from Greatest Element is Unique that $M_1 = M_2$.

That is, $(2)$ holds.

$\blacksquare$


Also see


Sources

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