Editing Incidence algebra (section)

==Special elements==
The multiplicative identity element of the incidence algebra is the '''[[Kronecker delta|delta function]]''', defined by

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

The '''zeta function''' of an incidence algebra is the constant function ''ζ''(''a'', ''b'') = 1 for every nonempty interval [''a, b'']. Multiplying by ''ζ'' is analogous to [[integral|integration]].

One can show that ζ is [[unit (ring theory)|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 inductively by the following relation:
:<math>\mu(x,y) = \begin{cases}
{}\qquad 1 & \text{if } x = y\\[6pt]
\displaystyle -\!\!\!\!\sum_{z\, :\,  x\,\leq\, z\, <\, y} \mu(x,z) & \text{for } x<y \\
{}\qquad 0 & \text{otherwise }.
\end{cases}</math>

Multiplying by ''μ'' is analogous to [[derivative|differentiation]], and is called [[Möbius inversion]].

The square of the zeta function gives the number of elements in an interval:
<math display="block">\zeta^2(x,y) = \sum_{z\in [x,y]} \zeta(x,z)\,\zeta(z,y)  = \sum_{z\in [x,y]} 1  =  \#[x,y].</math>