I have a few questions about a general partially ordered set which derive from a specific one.
$R$ is a partially ordered set over $A=\{a,b,c,d\}$ (4 different item group). $S=R\cup {(a,b)}$ is an equivalence relation.
I figured that $R$ mus be $\{(a,a),(b,b),(c,c),(d,d),(b,a)\}$, and I want to find the minimal and maximal elements.
Obviously not all of the elements are related so there are several but is $c$ and $d$ are BOTH minimal and maximal because they are alone?
You are correct. Since $c$ and $d$ are alone, $s\leq c$ and $c\leq s$ implies $s=c$, so $c$ is both minimal and maximal, and similarly $d$.
But there are more maximal and minimal elements. For example $a$ is maximal (but not minimal since $b\leq a$) and $b$ is minimal (not maximal for the same reason).