> Typically you would use "least" and "greatest" (for a partial order) or "first" and "last" (for a total order).
This is just absolutely not true. It is certainly "min" and "max" for any total order, they definitely don't have to be numbers or even numbers with the usual ordering (as confusing as that would be with a set of numbers but the reverse ordering). "first" and "last"? I've never heard that.
I haven't worked with partial orders so much that I can be as confident about that, perhaps "least" and "greatest" are the correct terms. But "min" and "max" are certainly used sometimes by non-specialists (i.e. professional mathmaticians that don't necessarily spend a lot of time with posets) without any confusion.
This is just absolutely not true. It is certainly "min" and "max" for any total order, they definitely don't have to be numbers or even numbers with the usual ordering (as confusing as that would be with a set of numbers but the reverse ordering). "first" and "last"? I've never heard that.
I haven't worked with partial orders so much that I can be as confident about that, perhaps "least" and "greatest" are the correct terms. But "min" and "max" are certainly used sometimes by non-specialists (i.e. professional mathmaticians that don't necessarily spend a lot of time with posets) without any confusion.