Special Elements
The multiplicative identity element of the incidence algebra is the delta function, defined by
The zeta function of an incidence algebra is the constant function ζ(a, b) = 1 for every interval . Multiplying by ζ is analogous to integration.
One can show that ζ 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) is invertible for every x.) The multiplicative inverse of the zeta function is the Möbius function μ(a, b); every value of μ(a, b) is an integral multiple of 1 in the base ring.
The Möbius function can also be defined directly, by the following relation:
Multiplying by μ is analogous to differentiation, and is called Möbius inversion.
Read more about this topic: Incidence Algebra
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