Definition:Greatest/Ordered Set

Definition
Let $\struct {S, \preceq}$ be an ordered set.

An element $x \in S$ is the greatest element (of $S$) :


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

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 $x$.

Also known as
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)
 * The unit element (in the context of boolean algebras and boolean rings)

Also see

 * Greatest Element is Unique
 * Greatest Element is Maximal
 * Maximal Element need not be Greatest Element


 * Definition:Smallest Element


 * Definition:Maximal Element
 * Definition:Minimal Element


 * Definition:Supremum of Set
 * Definition:Infimum of Set