The notion of minimum in Leinster's book

Question

On p.110 Leinster says that the minimum of $x,y\in \mathbb R$ satisfies $$\min\{x,y\}\le x,\ \min\{x,y\}\le y$$ and whenever $a\in\mathbb R$ satisfies $$a\le x, a\le y,$$ we have $a\le \min\{x,y\}$.

The last part looks counterintuitive to me. Isn't it supposed to say that whenever $a$ is less than or equal to both $x,y$, $a$ must be greater than or equal to $\min\{x,y\}$? Otherwise $a$ is "minimal", not $\min\{x,y\}$.

Also, is Leinster's notion the same as this definition (its version for minimum) on Wikipedia?

Answer 1

No, the statement is correct: the minimum of $x$ and $y$ in a linearly ordered set is also the infimum: the maximal lower bound for $\{x,y\}$ (which could exist in any partially ordered set with incomparable $x$ and $y$, see lattices etc.). Leinster is really saying that a linearly ordered set is a lattice. It might be (I don't know the book) that soon hereafter he goes on to define infimum, and this statement serves as motivation for that.

The statement $$a \le x, a \le y \implies a \le \min(x,y)$$ is trivial, as $\min(x,y)=x$ or $\min(x,y)=y$, depending on whether $x \le y$ or $x > y$ holds (and one of them must hold in a linear order).

Answer 2

Let $m$ be the minimum of $x,y$.

If $a \le m$, then $a\le m\le x,y$, so $a\le x,y$.

Conversely, suppose $a\le x,y$.

If $m < a$, then $m < a \le x,y$, so $m < x,y$, contradiction, since $m$ must be equal to one of $x,y$.

Thus, since we can't have $m < a$, it must be the case that $a\le m$.

Let's take a numerical example . . .

The minimum of $3,5$ is $3$, right?

Now suppose it's given that $a$ is a real number such that $a\le 3,5$.

Then $a\le 3$, so $a$ is less than or equal to the minimum of $3,5$.

It could be a lot less (e.g., we might have $a=-100$), but $a$ can't be more than the minimum.