The multiplicative identity element of the incidence algebra is
An incidence algebra is finitedimensional if and only if the underlying poset is finite.
The ζ function of an incidence algebra is the constant function ζ(a, b)=1 for every interval [a, b]. 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.) Its multiplicative inverse is the Möbius function μ(a, b); every value of μ(a, b) is an integral multiple of 1 in the base field.
In case the locally finite poset is the set of all positive integers ordered by divisibility, then its Möbius function is μ(a, b)=μ(b/a), where the second "μ" is the classic Möbius function introduced into number theory in the 19th century.
The locally finite poset of all finite subsets of some set E is ordered by inclusion. Here the Möbius function is
The Möbius function on the set of nonnegative integers with their usual order is
A poset is bounded if it has smallest and largest elements, which we call 0 and 1 respectively (not to be confused with the zero and the one of the base field, which, in this paragraph, we take to be Q). The Euler characteristic of a bounded finite poset is μ(0,1); it is always an integer. This concept is related to the classic Euler characteristic.
Incidence algebras of locally finite posets were treated in a number of papers of GianCarlo Rota beginning in 1964, and by many later combinatorialists.
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