I'm trying to find a poset with no smallest element, but with a unique minimal element.
I'm defining $a$ to be a minimal element with respect to the relation $\prec$ if $(x \prec a) \implies (x=a)$ and that $a$ is the smallest element if $a \prec x$ for all $x$.
This is what I have:
"Consider the the pair $(X, R)$, where $X=\mathbb{Z} \cup \{i\}$, and $R:= xRy \iff$ either $x,y \in \mathbb{Z}$ and $x<y$ or $x,y \in \{i\}$ and $x=y$.
Then $R$ is a partial ordering, and $X$ clearly has no smallest element.
Since no element of $\mathbb{Z}$ is comparable to $i$, then it must be that $i$ is the unique minimal element required."
Does this seem correct?