Definition:Maximal/Ordered Set/Comparison with Greatest Element

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Definition

Compare the definition of maximal element with that of a greatest element.


An element $x \in T$ is the greatest element of $T$ if and only if:

$\forall y \in T: y \preceq x$

That is, $x$ is comparable to, and succeeds, or is equal to, every $y \in S$.


An element $x \in S$ is a maximal element of $T$ if and only if:

$x \preceq y \implies x = y$

That is, $x$ succeeds, or is equal to, every $y \in S$ which is comparable to $x$.

If all elements are comparable to $x$, then such a maximal element is indeed the greatest element.


Note that when an ordered set is in fact a totally ordered set, the terms maximal element and greatest element are equivalent.


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