Editing Incidence algebra (section)

=== Natural numbers and ordinary generating functions ===
For the poset <math>(\mathbb{N},\leq),</math> the reduced incidence algebra consists of functions <math>f(a,b)</math> invariant under translation, <math>f(a+k,b+k) = f(a,b)</math> for all <math>k \ge 0,</math> so as to have the same value on isomorphic intervals [''a''+''k'', ''b''+''k''] and [''a'', ''b'']. Let ''t'' denote the function with ''t''(''a'', ''a''+1) = 1 and ''t''(''a'', ''b'') = 0 otherwise, a kind of invariant delta function on isomorphism classes of intervals. Its powers in the incidence algebra are the other invariant delta functions ''t''<sup>&thinsp;''n''</sup>(''a'', ''a''+''n'') = 1 and ''t''<sup>&thinsp;''n''</sup>(''x'', ''y'') = 0 otherwise. These form a [[basis (linear algebra)|basis]] for the reduced incidence algebra, and we may write any invariant function as <math>\textstyle f = \sum_{n\ge 0} f(0,n)t^n</math>. This notation makes clear the isomorphism between the reduced incidence algebra and the ring of formal power series <math>R[[t]]</math> over the scalars ''R,'' also known as the ring of ordinary [[generating function]]s. We may write the zeta function as <math>\zeta=1+t+t^2+\cdots = \tfrac1{1-t},</math> the reciprocal of the Möbius function <math>\mu=1-t.</math>