Definition:Smallest/Ordered Set

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

An element $x \in S$ is the smallest element :


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

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

The Smallest Element is Unique, so calling it the smallest element is justified.

The smallest element of $S$ is denoted $\min S$.

For an element to be the smallest element, all $y \in S$ must be comparable with $x$.

Also defined as
Some sources do not bother to define this concept on a general ordered set and instead apply it directly to a totally ordered set or even a well-ordered set.

Also known as
The smallest element of a set is also called:
 * The least element
 * The lowest element (particularly with numbers)
 * The first element
 * The minimum element (but beware confusing with minimal - see above)
 * The null element (in the context of boolean algebras and boolean rings)

Also see

 * Smallest Element is Unique


 * Definition:Greatest Element


 * Definition:Maximal Element
 * Definition:Minimal Element


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