# The covering relation

Warning: Tentative.

## Idea

The covering relation on a structure (generally already equipped with other relations) is a binary relation such that $x$ is related to $y$ if and only if $y$ is (in an appropriate sense) an immediate (and only immediate) successor of $x$.

## In a poset

A pair $\left(x,y\right)$ in a poset satsfies the covering relation if $x but there is no $z$ such that $x and $z. In other words, there are exactly two elements $z$ such that $x\le z\le y$: $z=x$ and $z=y$. In this case, you would say that ”$y$ covers $x$”.

## In a directed graph

A pair $\left(x,y\right)$ of vertices in a directed graph or quiver satisfies the covering relation if there is an edge $x\to y$ but there is no other path from $x$ to $y$.

## Common generalisation

Given any binary relation $\sim$ on a set $S$, a pair $\left(x,y\right)$ of elements of $S$ satisfies the covering relation if the only sequence $x={z}_{0},\dots ,{z}_{n}=y$ such that ${x}_{i}\sim {x}_{i+1}$ satisfies $n=1$ (so $x\sim y$). Then the covering relation on a poset is the covering relation of $\le$, and the covering relation in a directed graph is the covering relation of the adjacency relation of the graph.

## References

Revised on February 13, 2011 20:35:37 by Toby Bartels (75.88.68.70)