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  "body": "\n## Definition\n\nLet $(P, \\le)$ be a [[partially ordered set|0QV2JS9RJ3]] that is locally finite: for all $x \\le y$ in $P$, the interval $[x, y] = \\set{z \\in P : x \\le z \\le y}$ is finite.\nLet $K$ be a [[commutative ring|0310REJJY8]].\nThe *incidence algebra* $I(P, K)$ is the set of all functions $f \\colon \\set{(x, y) \\in P \\times P : x \\le y} \\to K$, with pointwise addition and multiplication by scalars, and with the product\n$$(f * g)(x, y) = \\sum_{z \\in [x, y]} f(x, z) \\, g(z, y).$$\n\n## Remarks\n\nThe product is associative, and the function $\\delta$ with $\\delta(x, x) = 1$ and $\\delta(x, y) = 0$ for $x \u003c y$ is the identity.\nThe zeta function is $\\zeta(x, y) = 1$ for all $x \\le y$.\nAn element $f$ is invertible if and only if $f(x, x)$ is a unit of $K$ for every $x$.\nThe inverse of $\\zeta$ is the Möbius function $\\mu$ of $P$, defined by $\\mu(x, x) = 1$ and $\\mu(x, y) = -\\sum_{x \\le z \u003c y} \\mu(x, z)$ for $x \u003c y$.\n\nExample: let $P = \\set{1, \\dots, n}$ with the usual order.\nEvery interval is finite, and $f \\mapsto (f(i, j))_{i \\le j}$ identifies $I(P, K)$ with the algebra of upper triangular $n \\times n$ matrices over $K$; the product $*$ becomes matrix multiplication.\n\nNon-example: $\\Q$ with its usual order is not locally finite, since $[0, 1]$ is infinite.\nThe sum defining $(\\zeta * \\zeta)(0, 1)$ would have infinitely many terms equal to $1$.\n",
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