Category:Smallest Elements

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This category contains results about Smallest Elements.
Definitions specific to this category can be found in Definitions/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 $S$ is denoted $\min S$.

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