# Confusion about definition of Maximum Element

I have some confusion about the definition of maximum elements. On the wikipedia page it says this:

Formally, given a partially ordered set (P, ≤), an element g of a subset S of P is the greatest element of S if

s ≤ g, for all elements s of S.


My question is: Why is the ≤ sign not a < sign? As I understand, a maximum element is unique within a poset (partially ordered set). It seems more logical to me that a < sign would be used instead of a ≤ sign, to ensure that the maximum is unique.

Is it because of the fact that in this poset (P, ≤) only the ≤ sign is relevant and that the < sign is not at play?

• Your last remark sounds fair, indeed. If I read it correctly, $g\in S$. Since $g\leq g$, but not $g<g$, I'd say $\leq$ would be more appropriate anyway. – HSN Nov 2 '17 at 16:52
• Correct; for PO-sets the symbol used is $\le$. – Mauro ALLEGRANZA Nov 2 '17 at 16:53

Assuming "$x<y$" to be equivalent to "$x\leq y\wedge x\neq y$", it should be clear that we cannot simply use that relation in the definition quoted above; if $x<g$ for all $x\in S$, then $g$ cannot also be in $S$ or else you get $g<g$, which is clearly false.
You can phrase the idea of a maximum alternate ways using "$<$", for instance $\forall x\in S(x\neq g\Rightarrow x<g)$; but the clause in parentheses there is the same thing as $x=g\vee x<g$, which is in turn the same thing as $x\leq g$, so we may as well just use the shorter one instead.
• Nothing. The sentence reads "for all $x$ in $S$, [statement]." – Malice Vidrine Nov 2 '17 at 23:59