# Given the numbers $x,y$ give an example of a partially ordered set with exactly $x$ minimal elements and exactly $y$ maximal elements

Given the numbers $x,y$ give an example of a partially ordered set with exactly $x$ minimal elements and exactly $y$ maximal elements

My idea is to employ the set of natural numbers and to extend it by some specific elements. Namely, consider this set: $$\mathbb N \cup \{m_1, \dots, m_x \} \cup \{M_1, \dots, M_y \}$$ Now, we use the standard $\le$ relation with a small nuance, namely: $m_i$ will be smaller than every natural number and $M_i$ will be greater than every natural number and for all $i, j$ neither $m_i \le m_j$ nor $M_i \le M_j$ and $m_i \le M_j$

What do you think of this? Can it work?

It can work, but why not make it even simpler and do away with $\mathbb N$? In other words, leave only $M$’s and $m$’s, making sure that all $M$’s are bigger than all $m$’s, but, between themselves, all $M$’s are incomparable, and so are the $m$’s.