Fence (mathematics)


In mathematics, a fence, also called a zigzag poset, is a partially ordered set in which the order relations form a path with alternating orientations:
or
A fence may be finite, or it may be formed by an infinite alternating sequence extending in both directions. The incidence posets of path graphs form examples of fences.
A linear extension of a fence is called an alternating permutation; André's problem of counting the number of different linear extensions has been studied since the 19th century. The solutions to this counting problem, the so-called Euler zigzag numbers or up/down numbers, are
The number of antichains in a fence is a Fibonacci number; the distributive lattice with this many elements, generated from a fence via Birkhoff's representation theorem, has as its graph the Fibonacci cube.
A partially ordered set is series-parallel if and only if it does not have four elements forming a fence.
Several authors have also investigated the number of order-preserving maps from fences to themselves, or to fences of other sizes.
An up-down poset Q is a generalization of a zigzag poset in which there are a downward orientations for every upward one and b total elements. For instance, Q has the elements and relations
In this notation, a fence is a partially ordered set of the form Q.

Equivalent conditions

The following conditions are equivalent for a poset P:
  1. P is a disjoint union of zigzag posets.
  2. If a ≤ b ≤ c in P, either a = b or b = c.
  3. < < =, i.e. it is never the case that a < b and b < c, so that < is vacuously transitive.
  4. P has dimension at most one.
  5. Every element of P is either maximal or minimal.
  6. The slice category Pos/P is cartesian closed.
The prime ideals of a commutative ring R, ordered by inclusion, satisfy the equivalent conditions above if and only if R has Krull dimension at most one.