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Sets can have minimal and least elements, and they are two different things, for example: given the set $A=\{\{1\},\{2\},\{3\},\{1,2\},\{2,3\},\{1,3\},\{1,2,3\}\}$ and the subset relation, this set has $\{1\}$ as a minimal element but not as a least element. Now, my question is, what is $\min(A)$? I would say that it doesn't exists since there are not least elements, but does $\min(A)$ really mean "least element of the set $A$"? I'm asking because the notation "$\min$" looks like it's referring to the minimal element and not the least one. And is this notation really referring to the least element of a set or instead it's a completely different things and we're talking about a unary operation used for the natural numbers, integers and real numbers?

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  • $\begingroup$ I assume you mean $a\lt b\iff a\subset b$. In which case this set has no minimum but has infimum $\{\}$. $\endgroup$ Commented Aug 11, 2020 at 9:23
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    $\begingroup$ I think that $\min$ is a short of "minimum", rather than of "minimal". $\endgroup$
    – Asaf Karagila
    Commented Aug 11, 2020 at 9:25
  • $\begingroup$ @AsafKaragila oh yeah of course! I was not thinking about it, least element and minimum element are the same things, thank you! $\endgroup$ Commented Aug 11, 2020 at 9:30

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Typically, $\min(A)$ is used in the context of a linear order (so all elements are comparable), and it means "minimum" (so the (unique) least element). I've never seen it used for "minimal element(s)" of a poset.

So your set $A$ has 3 minimal elements (the singletons), but no minimum indeed.

If you're using a textbook (or lecture notes of some kind), the notation $\min(A)$ should have been defined, but for me, when I see $\min(A)$ I assume implicitly that $A$ is linearly ordered and has a minimum.

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For all academic intent and purposes, Henno Brandsma's answer is the one I recommend. However, every time I get the opportunity to discuss this one peculiar natural correspondence between finitary sets and natural numbers I jump on it like a shark! It goes like this:

For any set, its integer equivalence is defined (meta-mathematically) by

$$S=\sum_{x\in S}2^x$$

(Note: $2^x$ is basic exponentiation, rather than the ambiguous notation for the power set.)

The reason I bring this up here is because, unlike the Von Neumann ordinals, it allows any finite set (of finite rank) to uniquely correspond to an integer. What this means for us is that all elements within $A$, can be rewritten into integers. Hence $\min(A)$ actually makes sense in this strange system.

$$A=\{\{1\},\{2\},\{3\},\{1,2\},\{2,3\},\{1,3\},\{1,2,3\}\}$$ $$\implies A=\{(2^1),(2^2),(2^3),(2^1+2^2),(2^2+2^3),(2^1+2^3),(2^1+2^2+2^3)\}$$ $$\implies A=\{2,4,8,6,12,10,14\}$$ $$\implies \min(A)=2=\{1\}$$

Application wise, this system draws it's roots from $p$-adic systems similar to the ones in Andrew Wile's famous proof of Fermat's Last theorem; so it does have some import.

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