Incidence Algebra - Special Elements

Special Elements

The multiplicative identity element of the incidence algebra is the delta function, defined by

\delta(a, b) = \begin{cases} 1 & \text{if } a=b \\ 0 & \text{if } a<b. \end{cases}

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:

\mu(x,y) = \begin{cases} {}\qquad 1 & \textrm{if}\quad x = y\\ \displaystyle -\sum_{z : x\leq z <y} \mu(x,z) & \textrm{for} \quad x<y \\ {}\qquad 0 & \textrm{otherwise}. \end{cases}

Multiplying by μ is analogous to differentiation, and is called Möbius inversion.

Read more about this topic:  Incidence Algebra

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