covering relation

Let $X$ be a set with a partial order $\backslash le$.

As usual, let be the relation on $X$ such that

Let $x$ and $y$ be elements of $X$.

Then $y$ covers $x$, written $x\backslash lessdot\; y$,

if

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

• In a finite linearly ordered set $\backslash \{1,\; 2,\; ...,\; n\backslash \}$, $i+1$ covers $i$ for all $i$ between $1$ and $n-1$, and there are no other covering relations.
• In the Boolean algebra of the power set of a set S, a subset B of S covers a subset A of S if and only if B is obtained from A by adding one element not in A.
• In Young's lattice, formed by the partitions of all nonnegative integers, a partition λ covers a partition μ if and only if the Young diagram of λ is obtained from the Young diagram of μ by adding an extra cell.
• The Hasse diagram depicting the covering relation of a Tamari lattice is the skeleton of an associahedron.
• The covering relation of any finite distributive lattice forms a median graph.
• On the real numbers with the usual total order ≤, no number covers another.

• If a partially ordered set is finite, its covering relation is the transitive reduction of the partial order relation. Such partially ordered sets are therefore completely described by their Hasse diagrams. On the other hand, in a dense order, such as the rational numbers with the standard order, no element covers another.

• {{Citation | last = Knuth | first = Donald E. | author-link = Donald Knuth | year = 2006 | title = The Art of Computer Programming, Volume 4, Fascicle 4 | publisher = Addison-Wesley | isbn = 0-321-33570-8}}.
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{{Order theory}}

Category:Binary relations

Category:Order theory