Seth Warner's Modern Algebra Exercise $14.22$ gets us exploring the properties of semilattices, in particular join semilattices.
Context: self-education.
The specific question that has been given is:
(b) If $(E, \vee)$ is a commutative idempotent semigroup such that every subset of $E$ is stable for $\vee$, then there is a unique total ordering $\le$ on $E$ such that $$x \vee y = \max \{ x, y\}$$
Notes:
- a commutative idempotent semigroup is better known as a semilattice, as I have learned from studies elsewhere;
- stable in this context means "closed": that is, every subset of $E$ is a subsemigroup of $E$.
It is straightforward though mildly laborious to prove that $\vee$ induces such a total ordering: reflexivity follows from idempotence, antisymmetry follows from commutativity, and transitivity from associativity, and connectedness follows from the closedness property as shown above.
Now, I've proved the existence of that total ordering, and I have shown that it has the property $x \vee y = \max \{ x, y\}$.
Basically, the ordering $\le$ is defined as: $$x \le y \iff x \vee y = y$$
and the rest follows easily.
Now, what I don't understand how to do is that final bit:
prove that $\le$ is a unique total ordering.
The only way I can conceive of a total ordering being "unique" is by showing that: suppose $\le_1$ and $\le_2$ are total orderings.
Then $\le_1$ and $\le_2$ are the same total ordering iff for any arbitrary $x \in E$, we have:
- $\{y \in E: y \le_1 x\} = \{y \in E: y \le_2 x\}$
- $\{z \in E: x \le_1 z\} = \{z \in E: x \le_2 z\}$
which looks horribly daunting to prove for the general set $E$, and I really don't know where to start.
You are supposed to prove that there is a unique total ordering with the property that $\max(x,y)$ (defined in terms of the ordering) is always equal to $x\vee y$. In other words, you want to show that if $\leq'$ is any other total ordering with this property, it must be equal to your $\leq$ defined by $x\leq y$ iff $x\vee y=y$. But this is trivial, since by assumption $\max_{\leq'}(x,y)=x\vee y$ and $\max_{\leq'}(x,y)=y$ iff $x\leq'y$.