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A poset ''P'' is said to be a differential poset, and in particular to be ''r''-differential (where ''r'' is a positive integer), if it satisfies the following conditions:
* ''P'' is [[graded poset|graded]] and [[locally finite poset|locally finite]] with a unique minimal element;
* for every two distinct elements ''x'', ''y'' of ''P'', the number of elements [[covering relation|covering]] both ''x'' and ''y'' is the same as the number of elements covered by both ''x'' and
* for every element ''x'' of ''P'', the number of elements covering ''x'' is exactly ''r'' more than the number of elements covered by
These basic properties may be restated in various ways. For example, Stanley shows that the number of elements covering two distinct elements ''x'' and ''y'' of a differential poset is always either 0 or 1, so the second defining property could be altered accordingly.
The defining properties may also be restated in the following [[linear algebra]]ic setting: taking the elements of the poset ''P'' to be formal [[basis (linear algebra)|basis]] vectors of an (infinite dimensional) [[vector space]], let ''D'' and ''U'' be the [[linear operator|operators]] defined so that ''D'' ''x'' is equal to the sum of the elements covered by ''x'', and ''U'' ''x'' is equal to the sum of the elements covering
This latter reformulation makes a differential poset into a combinatorial realization of a [[Weyl algebra]], and in particular explains the name ''differential'': the operators "''d''/''dx''" and "multiplication by ''x''" on the vector space of polynomials obey the same commutation relation as ''U'' and ''D''/''r''.
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