Covering relation

In mathematics, especially order theory, the covering relation of a partially ordered set is the binary relation which holds between comparable elements that are immediate neighbours. The covering relation is commonly used to graphically express the partial order by means of the Hasse diagram.

Let ${\displaystyle X}$ be a set with a partial order ${\displaystyle \leq }$. As usual, let ${\displaystyle <}$ be the relation on ${\displaystyle X}$ such that ${\displaystyle x<y}$ if and only if ${\displaystyle x\leq y}$ and ${\displaystyle x\neq y}$.

Let ${\displaystyle x}$ and ${\displaystyle y}$ be elements of ${\displaystyle X}$.

Then ${\displaystyle y}$ covers ${\displaystyle x}$, written ${\displaystyle x\lessdot y}$, if ${\displaystyle x<y}$ and there is no element ${\displaystyle z}$ such that ${\displaystyle x<z<y}$. Equivalently, ${\displaystyle y}$ covers ${\displaystyle x}$ if the interval ${\displaystyle [x,y]}$ is the two-element set ${\displaystyle \{x,y\}}$. In more intuitive words, ${\displaystyle x\lessdot y}$ if ${\displaystyle y}$ immediately supersedes or succeeds ${\displaystyle x}$ in terms of their respective poset's order relation.

When ${\displaystyle x\lessdot y}$, it is said that ${\displaystyle y}$ is a cover of ${\displaystyle x}$. Some authors also use the term cover to denote any such pair ${\displaystyle (x,y)}$ in the covering relation.