Since strict partial orders are irreflexive, antisymmetric and transitive, I am confused as to how they can have minimal/maximal elements as surely these would be undefined due to them not being able to have terms that are the same?
Can we actually define minimal/maximal definitions for strict partial orders?
Additionally, is the definition for least/greatest for a strict partial order as follows:
a is least is defined as for all b in a, a < b
a is greatest is defined as for all b in a, b < a
Does this need any other conditions?
Thank you!
$a$ is the maximum (greatest element) of a set $A$ if $a \in A$ and for all $b \in A$ we have $b=a$ or $a < b$.
The minimum (least element of $A$) is defined analogously. These need not exist.
$a \in A$ is a maximal element of $A$ iff no $b \in A$ exists such that $a < b$.
A minimal element is again defined analogously. A maximum is maximal, but the reverse need not hold etc.