maximal and minimal elements

Distinguishing elements of a set equipped with a preorder:

A maximal element is not less than any other element of the set.
A minimal element is not greater than any other element of the set.

Formally,
for a preorder $\le$ over set $S$,

• an element $M\in S$ is a maximal element of $S$ if whenever $M\le s$, then $s\le M$
  • equivalently, $s\le M$ whenever $M,s$ are comparable
• an element $m\in S$ is a minimal element of $S$ if whenever $s\le m$, then $m\le s$
  • equivalently, $m\le s$ whenever $m,s$ are comparable

N.B. it is not required that a maximal or minimal element is comparable to every other element.

If $S$ is equipped with a partial order,

• an element $M\in S$ is a maximal element of $S$ if there is no $s\in S$ such that $M\le s$ with $s\ne M$.
• an element $m\in S$ is a minimal element of $S$ if there is no $s\in S$ such that $s\le m$ with $s\ne m$.