Incidence algebra

In order theory, a field of mathematics, a locally finite partially ordered set is one for which every closed interval

Examples

• 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 finite subsets of some set E, ordered by inclusion, form a locally finite poset. Here the Möbius function is

::mu(S,T)=(-1)^{left|Tsetminus S ight|}

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

:whenever S and T are finite subsets of E with S ⊆ T.

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

• The Möbius function on the set of non-negative integers with their usual order is

::mu(x,y)=left{egin{matrix}

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

1 & mbox{if }y-x=0, \

~ ~ ~ ~ ~ ~ ~ ~ ~ ~