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  "body": "\n## Summary\n\nThe incidence algebra of a locally finite [[partially ordered set|0QV2JS9RJ3]] is the algebra of functions on its intervals, multiplied by a convolution.\nIts most important elements are the zeta function, which is $1$ on every interval, and its inverse, the Möbius function of the poset.\nIt gives a common setting for inclusion–exclusion, the number-theoretic [[Möbius inversion formula|07Q4TBTFVB]] and the calculus of finite differences, through [[Möbius inversion on posets|B8XJSJ517Q]].\n\n## Explanations\n\n### L1 human\n\nTake a poset in which every interval $[x, y]$ has finitely many elements.\nAttach a number to each pair $x \\le y$, and multiply two such functions by summing over all the points $z$ between $x$ and $y$.\nThis is matrix multiplication in disguise.\nFor the chain $1 \u003c 2 \u003c \\dots \u003c n$ the incidence algebra is the algebra of upper triangular $n \\times n$ matrices.\n",
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