A finite poset is said to be a fence if it has a Hasse diagram that looks like one of the following four types:

1. \begin{equation*} \xymatrix @C-=2pt @R=6pt { & & b_1 \ar@{-}[lldd] \ar@{-}[rrdd] & & & & b_2 \ar@{-}[lldd] \ar@{-}[rd] \\ & & & & & & & \\ a_1 & & & & a_2 & & } \xymatrix @C-=2pt @R=6pt { {} \\ {\cdots} \\ } \xymatrix @C-=2pt @R=6pt { & & & b_{m-1} \ar@{-}[lldd] \ar@{-}[rrdd] & & & & b_m \ar@{-}[lldd] \\ \ar@{-}[rd] & & & & & & & & \\ & a_{m-1} & & & & a_m & & } \end{equation*}
2. \begin{equation*} \xymatrix @C-=2pt @R=6pt { b_1 \ar@{-}[rrdd] & & & & b_2 \ar@{-}[lldd] \ar@{-}[rrdd] & & \\ & & & & & & & \ar@{-}[ld] \\ & & a_1 & & & & a_2 } \xymatrix @C-=2pt @R=6pt { {} \\ {\cdots} \\ } \xymatrix @C-=2pt @R=6pt { & b_{n-1} \ar@{-}[ld] \ar@{-}[rrdd] & & & & b_n \ar@{-}[lldd] \ar@{-}[rrdd] & & \\ & & & & & & & & \\ & & & a_{n-1} & & & & a_n } \end{equation*}
3. \begin{equation*} \xymatrix @C-=2pt @R=6pt { & & b_1 \ar@{-}[lldd] \ar@{-}[rrdd] & & & & b_2 \ar@{-}[lldd] \ar@{-}[rd] \\ & & & & & & & \\ a_1 & & & & a_2 & & } \xymatrix @C-=2pt @R=6pt { {} \\ {\cdots} \\ } \xymatrix @C-=2pt @R=6pt { & & & b_{p-1} \ar@{-}[lldd] \ar@{-}[rrdd] & & & & b_p \ar@{-}[lldd] \ar@{-}[rrdd] & & \\ \ar@{-}[rd] & & & & & & & & & & \\ & a_{p-1} & & & & a_p & & & & a_{p+1}} \end{equation*}
4. \begin{equation*} \xymatrix @C-=2pt @R=6pt { b_1 \ar@{-}[rrdd] & & & & b_2 \ar@{-}[lldd] \ar@{-}[rrdd] & & \\ & & & & & & & \ar@{-}[ld] \\ & & a_1 & & & & a_2 } \xymatrix @C-=2pt @R=6pt { {} \\ {\cdots} \\ } \xymatrix @C-=2pt @R=6pt { & b_{q-1} \ar@{-}[ld] \ar@{-}[rrdd] & & & & b_q \ar@{-}[lldd] \ar@{-}[rrdd] & & & & b_{q+1} \ar@{-}[lldd] \\ & & & & & & & & & & \\ & & & a_{q-1} & & & & a_q & &} \end{equation*}

When no pairs of $m,n,p,q$ are the same, no two types are order isomorphic. A fence of type 1 or type 2 are called an even fence, for the obvious reason that it contains an even number of elements. The other two types are the odd fences.

When $m=n$ the two even fences are isomorphic, simply by mapping $c_i$ in the first fence to $c_{n-i}$ in the second one, where $c$ is either $a$ or $b$ When $p=q$ the two odd fences are dually isomorphic to one another (the dual of one fence is isomorphic to the other fence).

Another property of a fence is that it is connected and is minimal in the sense that, among all partial orderings on the underlying set, a fence has the least number of elements in its partial ordering. Of course, the converse is not true, as the following example illustrates \begin{equation*} \xymatrix { b_1 \ar@{-}[rrd] & b_2 \ar@{-}[rd] & \cdots & b_{n-1} \ar@{-}[ld] & b_n \ar@{-}[lld] \\ & & a_1 & & } \end{equation*} It can be shown that the number of lower sets in a fence is a Fibonacci number. In other words, if $f(n)$ is the number of lower sets in a fence of cardinality $n$ ($n$ at least $2$ , then $f(n+1)=f(n)+f(n-1)$ and $f(2)=2$

Remark. One may extend the definition of a fence to the infinitely countable case. In this case, it has a Hasse diagram that looks like \begin{equation*} \xymatrix { {} \ar@{}[d]_-{\displaystyle{\stackrel{}{\cdots}}} \\ {} } \xymatrix { \ar@{-}[rd] & & b_{n-1} \ar@{-}[rd] \ar@{-}[ld] & & b_n \ar@{-}[ld] \ar@{-}[rd] & & b_{n+1} \ar@{-}[ld] \ar@{-}[rd] & \\ & a_{n-1} & & a_n & & a_{n+1} & & } \xymatrix { {} \ar@{}[d]_-{\displaystyle{\stackrel{}{\cdots}}} \\ {} } \end{equation*}Its underlying set is the disjoint union of the two countably infinite sets $A=\lbrace a_i\mid i\in \mathbb{Z}\rbrace$ and $B=\lbrace b_i\mid i\in \mathbb{Z}\rbrace$ and its partial order is the disjoint union of the three sets $\lbrace (a_i,b_i)\mid i\in \mathbb{Z}\rbrace$ $\lbrace (a_i,b_{i-1})\mid i\in \mathbb{Z}\rbrace$ and $\lbrace (c,c)\mid c\in A\cup B\rbrace$

Bibliography

1

B. A. Davey, H. A. Priestley, Introduction to Lattices and Order, 2nd Edition, Cambridge (2003)