Infimum and supremum of the empty set

There might be different preferences to how one should define this. I am not sure that I understand exactly what you are asking, but maybe the following can be helpful.

If we consider subsets of the real numbers, then it is customary to define the infimum of the empty set as being $\infty$. This makes sense since the infimum is the greatest lower bound and every real number is a lower bound. So $\infty$ could be thought of as the greatest such.

The supremum of the empty set is $-\infty$. Again this makes sense since the supremum is the least upper bound. Any real number is an upper bound, so $-\infty$ would be the least.

Note that when talking about supremum and infimum, one has to start with a partially ordered set $(P, \leq)$. That $(P, \leq)$ is partial ordered means that $\leq$ is reflexive, antisymmetric, and transitive.

So let $P = \mathbb{R} \cup \{-\infty, \infty\}$. Define $\leq$ the "obvious way", so that $a\leq \infty$ for all $a\in \mathbb{R}$ and $-\infty \leq a$ for all $a\in \mathbb{R}$.

With this definition you have a partial order and it this setup the infimum and the supremum are as mentioned above. So you don't need non-empty.

When you say "meaningful" you actually mean to say something "$\sup(E)$ exists". But what does it mean to exist? In the context of the real numbers it means that there exists a real number $r$ such that $r=\sup(E)$.

Of course $\pm\infty$ are not real numbers, but they still contain information for us. When we say that $\sup(E)=+\infty$ then we say that $E$ does not have a supermum in the real numbers, or equivalently for every $r\in\Bbb R$ we can find $e\in E$ such that $r<e$.

So in fact we can extend the definition of $\sup$ and $\inf$ to include $\pm\infty$ as possible values by adding two formal symbols for these "points at infinity".

Now, what was the definition of $\sup(E)$ anyway? It was the least point which is not smaller than any of the points in $E$. That is, $\sup(E)=x\iff\forall r\in E:r\leq x\land\forall z(\forall r\in E(r\leq z)\rightarrow x\leq z)$.

Plug this to the empty set, then $\forall r\in\varnothing$ is vacuously true, so every $r\in\Bbb R$ satisfies this condition. It follows that $-\infty$ also satisfies this condition. Therefore $\sup(\varnothing)=-\infty$.

If grasping this is somewhat difficult, here is an equivalent way of looking at this. Consider only $[0,1]$, and now define $\sup^*$ and $\inf^*$ as before but for subsets of $[0,1]$ and requiring that the $\sup^*$ and $\inf^*$ are from $[0,1]$ as well. What is $\sup^*(\varnothing)$?

Let $\Bbb{R}$ be the set of real numbers with the usual order and let $\overline{\Bbb{R}}=\Bbb{R}\cup\{+\infty;-\infty\}$ with the following rules:$$-\infty< x< +\infty~~\forall x\in\Bbb{R}$$ Let $E=\emptyset$ be a subset of $\overline{\Bbb{R}}$. $U$ is an upper bound of $E$ if: $$\exists U\in\Bbb{R};~x\leq U~~~\forall x\in E$$ and $E$ doesn't have an upper bound in $\Bbb{R}$ if:$$\forall U\in\Bbb{R},~\exists x\in E;~~~x>U$$ which is clearly false since $E$ doesn't contain any element, so $E$ doesn't have an upper bound in $\Bbb{R}$. Let us define $\sup(E)=-\infty$ (you'll see why in a moment).

We have the relation $\sup(A\cup B)=\max\{\sup(A);\ \sup(B)\}$. If $B=\emptyset$, we expect $\sup(A\cup B)$ to be equal to $\sup(A)$ which is only verified if $\sup(\emptyset)=-\infty$

It seems you are aware of the following:

Fact: For any non-empty set of real numbers $E$ with an upper (lower) bound in $\Bbb R$, there is a least (greatest) upper (lower) bound. We call this $\sup E$ ($\inf E$).

Now, suppose we are given any set $E\subseteq R$. Define $U(E):=\{x\in\Bbb R:\forall y\in E(y\le x)\}$ to be the set of upper bounds of $E$ in $\Bbb R$. Similarly, let $L(E):=\{x\in\Bbb R:\forall y\in E(x\le y)\}$ be the set of lower bounds of $E$. You should be able to show the following:

• $U(E)$ is an upward-closed set--that is, if $x\in U(E)$ and $z\in\Bbb R$ with $x\le z,$ then $z\in U(E)$. Likewise, $L(E)$ is downward-closed.
• The only upward-closed subsets of $\Bbb R$ are $\emptyset,$ $\Bbb R,$ and the sets of the form $(a,\rightarrow):=\{x\in\Bbb R:a<x\}$ and $[a,\rightarrow):=\{x\in\Bbb R:a\le x\}$ for $a\in\Bbb R$. Similarly, the downward-closed subsets of $\Bbb R$ are $\emptyset,$ $\Bbb R,$ and the sets $(\leftarrow,b):=\{x\in\Bbb R:x<b\}$ and $(\leftarrow,b]:=\{x\in\Bbb R:x\le b\}.$
• $\Bbb R$ is the only upward-closed (downward-closed) subset of reals that is unbounded below (above).
• $E=\emptyset$ if and only if $U(E)=L(E)=\Bbb R.$

Now we can see the following:

Fact (rephrased): For any non-empty set of real numbers $E,$ either $U(E)$ is empty--meaning that $E$ has no upper bound in $\Bbb R$--or $U(E)$ has a least element--meaning that $U(E)=[a,\rightarrow)$ for some $a\in\Bbb R.$ This $a$ is uniquely determined by $E$, and we call it $\sup E$. Likewise, for any non-empty set of real numbers $E,$ either $L(E)$ is empty or $L(E)=(\leftarrow,b]$ for some unique $b\in\Bbb R$, which we call $\inf E$.

In the case that $E=\emptyset$, we have $U(E)=\Bbb R,$ which has no least element, so we can't say that there is any real number that we call $\sup E$ in this case. Accordingly, "$\sup E=-\infty$" should be interpreted (using my notation) to mean that $U(E)$ is unbounded below, which (by the discussion in the bullet points) is just another way of saying that $E$ is empty. Likewise, "$\inf E=+\infty$" means that $L(E)$ is unbounded above, just another way of saying that $E$ is empty.

Finally, note that if $E$ is non-empty and bounded above, then $\sup E=\inf U(E)$. In this light, we should interpret "$\sup E=+\infty$" in the same way as "$\inf U(E)=+\infty$," meaning $U(E)$ is empty (see previous paragraph), meaning that $E$ is unbounded above. Similarly, we should interpret "$\inf E=-\infty$" as saying that $E$ is unbounded below. These conventions are usually easier to grasp/swallow than that "$\sup E=-\infty$" and "$\inf E=+\infty$" both mean "$E=\emptyset$"--at least, that has been my experience--and I know you didn't directly ask about them, but I felt they were worth mentioning for elucidation.