I am working through a problem inspired by this unanswered question:
Help understanding a proof about finite lattice
The hint from the book says "let $b$ be the lower cover of $a$ in the chain $C$".
What do they mean by "the lower cover of a"? Do they mean the element above $a$ in the chain? Or is there a definition of "lower cover" that I am unaware of?
Definition (Schröder 2016): For elements $a, b$ in a poset, $a$ is a lower cover of $b$ iff $a < b$ and no element $c$ satisfies $a < c < b$.
So for a non-minimal element $a$ in a finite chain, there is a unique lower cover of $a$.