Category:Definitions/Greatest Elements

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

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

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

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

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 $x$.