Let $A = \{a,b,c\}$. Use Hasse diagrams to describe all partial orderings on $A$ for which $a$ is a minimal element (there should be 10 partial orderings I guess). I know how to draw Hasse diagrams for simple posets, but this problem became relatively harder for me.
2026-04-18 23:39:15.1776555555
Hasse diagram for minimal element
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1
There are ten, indeed.
I will not draw the posets, but I'll describe them to you in a way that must be easy to follow.
(1) There is one anti-chain.
(4) There are four poset consisting of a two-element chain and another element which is not connected to neither of those: in two of them, $a$ is both minimal and maximal (one with $b$ over $c$, and the other with $c$ over $b$); in the other two, one of them has $b$ over $a$, and the other has $c$ over $a$.
(2) There are two wedge-shaped posets ($\wedge$): one with $b$ on top ($a$ and $c$ on bottom) and another with $c$ on top.
(1) There's a v-shaped one, with $a$ in the bottom.
(2) There are two chains: one with $b$ on the top, and the other with $c$ on the top.
$1+4+2+1+2=10$.