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A finite poset is said to be a fence if it has a Hasse diagram that looks like one of the following four types:
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where  are positive integers.
When no pairs of  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  , the two even fences are isomorphic, simply by mapping  in the first fence to  in the second one, where  is either  or  . When  , 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
It can be shown that the number of lower sets in a fence is a Fibonacci number. In other words, if  is the number of lower sets in a fence of cardinality  ( at least  ), then
 and  .
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
Its underlying set is the disjoint union of the two countably infinite sets
 and
 , and its partial order is the disjoint union of the three sets
 ,
 , and
 .
- 1
- B. A. Davey, H. A. Priestley, Introduction to Lattices and Order, 2nd Edition, Cambridge (2003)
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"fence" is owned by CWoo. [ full author list (2) ]
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(view preamble)
| Also defines: |
even fence, odd fence |
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Cross-references: countably infinite, disjoint union, countable, cardinality, Fibonacci number, lower sets, converse, least number, minimal, property, mapping, even number, contains, obvious, isomorphic, order, integers, positive, types, Hasse diagram, poset, finite
There is 1 reference to this entry.
This is version 4 of fence, born on 2007-05-22, modified 2007-05-23.
Object id is 9439, canonical name is Fence.
Accessed 841 times total.
Classification:
| AMS MSC: | 06A06 (Order, lattices, ordered algebraic structures :: Ordered sets :: Partial order, general) |
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Pending Errata and Addenda
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