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

Famous quotes containing the word special:

    “People generally will soon understand that writers should be judged, not according to rules and species, which are contrary to nature and art, but according to the immutable principles of the art of composition, and the special laws of their individual temperaments.”
    —Victor Hugo (1802–1885)