Editing Incidence algebra (section)

==Examples==
;'''Positive integers ordered by divisibility'''
:The convolution associated to the incidence algebra for intervals [1, ''n''] becomes the [[Dirichlet convolution]], hence the Möbius function is ''μ''(''a, b'') = ''μ''(''b/a''), where the second "''μ''" is the classical [[Möbius function]] introduced into number theory in the 19th century.
;'''Finite subsets of some set ''E'', ordered by inclusion'''
:The Möbius function is <math display="block">\mu(S,T)=(-1)^{\left|T\smallsetminus S\right|}</math>
:whenever ''S'' and ''T'' are finite [[subset]]s of ''E'' with ''S'' ⊆ ''T'', and Möbius inversion is called the [[principle of inclusion-exclusion]].
:Geometrically, this is a [[hypercube]]: <math>2^E = \{0,1\}^E.</math>
;'''Natural numbers with their usual order'''
:The Möbius function is <math display="block">\mu(x,y) = \begin{cases}
1& \text{if }y=x, \\
-1 & \text{if }y = x+1, \\
0 & \text{if }y>x+1,
\end{cases}
</math> and Möbius inversion is called the (backwards) [[difference operator]].
:Geometrically, this corresponds to the discrete [[number line]].
:The convolution of functions in the incidence algebra corresponds to multiplication of [[formal power series]]: see the discussion of reduced incidence algebras below. The Möbius function corresponds to the sequence (1, −1, 0, 0, 0, ... ) of coefficients of the formal power series 1&thinsp;−&nbsp;''t'', and the zeta function corresponds to the sequence of coefficients (1, 1, 1, 1, ...) of the formal power series <math>(1 - t)^{-1} = 1 + t + t^2 + t^3 + \cdots</math>, which is inverse. The delta function in this incidence algebra similarly corresponds to the formal power series 1.
;'''Finite sub-multisets of some multiset ''E'', ordered by inclusion'''
:The above three examples can be unified and generalized by considering a [[multiset]] ''E,'' and finite sub-multisets ''S'' and ''T'' of ''E''. The Möbius function is <math display="block"> \mu(S,T) = \begin{cases}
0 & \text{if } T \smallsetminus S \text{ is a proper multiset (has repeated elements)}\\
(-1)^{\left|T \smallsetminus S\right|} & \text{if } T\smallsetminus S \text{ is a set (has no repeated elements)}.
\end{cases}</math>
:This generalizes the positive [[integer]]s ordered by [[divisibility]] by a positive integer corresponding to its multiset of [[prime number|prime]] [[divisor|factors]] with multiplicity, e.g., 12 corresponds to the multiset <math>\{ 2, 2, 3 \}.</math>
:This generalizes the [[natural number]]s with their usual order by a natural number corresponding to a multiset of one underlying element and cardinality equal to that number, e.g., 3 corresponds to the multiset <math>\{ 1, 1, 1 \}.</math>
;'''Subgroups of a finite [[p-group|''p''-group]] ''G'', ordered by inclusion'''
:The Möbius function is <math display="block">\mu_G(H_1,H_2) = (-1)^{k} p^{\binom{k}{2}}</math> if <math>H_1</math> is a [[normal subgroup]] of <math>H_2</math> and <math>H_2/H_1 \cong (\Z/p\Z)^k</math> and it is 0 otherwise. This is a theorem of Weisner (1935).
;'''Partitions of a set'''
:Partially order the set of all [[partition of a set|partitions]] of a finite set by saying ''σ'' &le; ''τ'' if ''σ'' is a finer partition than ''τ''.  In particular, let ''τ'' have ''t'' blocks which respectively split into ''s''<sub>1</sub>, ..., ''s''<sub>''t''</sub> finer blocks of ''σ'', which has a total of ''s'' = ''s''<sub>1</sub> + &sdot;&sdot;&sdot; + ''s''<sub>''t''</sub> blocks. Then the Möbius function is: <math display="block">\mu(\sigma,\tau) =
(-1)^{s-t}(s_1{-}1)! \dots (s_t{-}1)!</math>