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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. 

Link: https://en.wikipedia.org/wiki/Greatest_and_least_elements

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?

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  • $\begingroup$ 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. $\endgroup$ – HSN Nov 2 '17 at 16:52
  • $\begingroup$ Correct; for PO-sets the symbol used is $\le$. $\endgroup$ – Mauro ALLEGRANZA Nov 2 '17 at 16:53
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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.

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  • $\begingroup$ What does S(statement) mean? $\endgroup$ – William Elliot Nov 2 '17 at 21:10
  • $\begingroup$ Nothing. The sentence reads "for all $x$ in $S$, [statement]." $\endgroup$ – Malice Vidrine Nov 2 '17 at 23:59
  • $\begingroup$ It does not. Contrary to your remark, it says for all x in S(statement). $\endgroup$ – William Elliot Nov 3 '17 at 2:08
  • $\begingroup$ Conventions regarding how to denote bounded quantifiers and their scope is uneven, and what I have done is not the prettiest way, but it's not an uncommon way. In any case, any future readers have had the situation disambiguated, if it wasn't already clear by the alternative but making sense in context. $\endgroup$ – Malice Vidrine Nov 3 '17 at 4:08
  • $\begingroup$ A comma after S is expected. $\endgroup$ – William Elliot Nov 3 '17 at 7:44

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