Incidence algebra

In order theory, a field of mathematics, a locally finite partially ordered set is one for which every closed interval

Related Topics:
Order theory - Mathematics - Partially ordered set - Interval

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: = {x : a ≤ x ≤ b}

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within it is finite. For every locally finite poset and every field of scalars there is an incidence algebra, an associative algebra defined as follows. The members of the incidence algebra are the functions f assigning to each interval a scalar f(a, b). On this underlying set one defines addition and scalar multiplication pointwise, and "multiplication" in the incidence algebra is a convolution defined by

Related Topics:
Finite - Field - Scalar - Associative algebra - Function - Convolution

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:(f*g)(a, b)=sum_{aleq xleq b}f(a, x)g(x, b).

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The multiplicative identity element of the incidence algebra is

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:

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delta(a, b) = left{

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egin{matrix}

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,1, & mbox{if } a=b \

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,0, & mbox{if } a

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end{matrix}

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ight.

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An incidence algebra is finite-dimensional if and only if the underlying partially ordered set is finite.

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The ζ function of an incidence algebra is the constant function ζ(a, b) = 1 for every interval . One can show that that element is invertible in the incidence algebra (with respect to the convolution defined above). (Generally, a member h of the incidence algebra is invertible if and only if h(x, x) ≠ 0 for every x.) The multiplicative inverse of the ζ function is the Möbius function μ(a, b); every value of μ(a, b) is an integral multiple of 1 in the base field.

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