Checking if the mimimum is unique
You cannot avoid enumerating all elements. Consider the following two posets, with elements $x,y,z_1,\ldots,z_n$:


$x,y &lt; z_i$ for all $i$.
$x,y &lt; z_i$ for all $i &lt; n$, and $z_n &lt; x,y$.


The first poset has two minima, and second only one. You can only tell the difference by enumerating over all elements.



If there is a unique minimal element $z$, then $z &lt; w$ for all other elements $w$ (since there is a minimal element below any given element). This allows us to use the following constant space algorithm, which assumes that the elements of the poset are $z_1,\ldots,z_n$:


Initialize $t \gets z_1$.
For $i=2,\ldots,n$: if $z_i &lt; t$, let $t \gets z_i$.
Go over all elements again, and verify that $t \leq z_i$ for all of them.


There is a unique minimal element iff the verification succeeds.