Definition:Greatest Element

Definition
Let $\left({S, \preceq}\right)$ be a poset.

An element $x \in S$ is the greatest element iff:


 * $\forall y \in S: y \preceq x$

That is, every element of $S$ precedes, or is equal to, $x$.

The Greatest Element is Unique, so calling it the greatest element is justified.

Thus for an element $x$ to be the greatest element, all $y \in S$ must be comparable to $x$.

Comparison with Maximal Element
Compare this definition with that for a maximal element.

An element $x \in S$ is maximal iff:


 * $x \preceq y \implies x = y$

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

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

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

Alternative Names
The greatest element of a set is also called:
 * The largest element (or biggest element, etc.)
 * The last element
 * The maximum element (but beware confusing with maximal - see above)

Also see

 * Greatest Element is Unique


 * Smallest element


 * Maximal element
 * Minimal element


 * Supremum of an Ordered Set
 * Infimum of an Ordered Set