nLab 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 $(x,y)$ in a poset satisfies the covering relation if $x \lt y$ but there is no $z$ such that $x \lt z$ and $z \lt y$. In other words, the interval $[x,y] = \{z \mid x \leq z \leq y\}$ contains exactly two elements $x$ and $y$. In this case, you would say that “$y$ covers $x$”.

In a directed graph

A pair $(x,y)$ 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 $(x,y)$ of elements of $S$ satisfies the covering relation if the only sequence $x = z_0, \ldots, z_n = y$ 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 $\leq$, and the covering relation in a directed graph is the covering relation of the adjacency relation of the graph.

References

• Wikipedia