Definition:Greatest/Ordered Set

Definition

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

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

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)

Also see

• Greatest Element is Unique
• Greatest Element is Maximal
• Maximal Element Not Necessarily Greatest Element

• Smallest element

• Maximal element
• Minimal element

• Supremum of an Ordered Set
• Infimum of an Ordered Set

Sources

• Paul R. Halmos: Naive Set Theory (1960)... (previous)... (next): $\S 14$: Order
• T.S. Blyth: Set Theory and Abstract Algebra (1975): $\S 7$
• K.G. Binmore: Mathematical Analysis: A Straightforward Approach (1977)... (previous)... (next): $\S 2.7$