Category:Definitions/Smallest Elements

This category contains definitions related to Smallest Elements.
Related results can be found in Category:Smallest Elements.

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

An element $x \in S$ is the smallest element if and only if:

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

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 $S$ can be denoted:

$\map \min S$

$0$

$\mathrm O$

or similar.

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