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  "body": "\n## Definition\n\nLet $(I, \\le)$ be a directed partially ordered set and fix a signature.\nAn *inverse system* of algebras over $I$ consists of:\n\n1. an algebra $A_i$ for each $i \\in I$;\n2. a [[homomorphism|KBXSZC6BV1]] $\\varphi_{ij} \\colon A_j \\to A_i$ for each pair $i \\le j$;\n3. the conditions $\\varphi_{ii} = \\id_{A_i}$ and $\\varphi_{ij} \\circ \\varphi_{jk} = \\varphi_{ik}$ for all $i \\le j \\le k$.\n\nThe *inverse limit* $\\varprojlim A_i$ of the system is the set of all $(a_i)_{i \\in I}$ in the [[direct product|WJ73XYBNE4]] $\\prod_{i \\in I} A_i$ such that $\\varphi_{ij}(a_j) = a_i$ for all $i \\le j$.\n\n## Remarks\n\nThe inverse limit is closed under the operations of the product, so it is a [[subalgebra|7CHG6SD365]] of $\\prod_i A_i$; it can be empty when the signature has no constant symbols.\nThe projections $\\pi_i \\colon \\varprojlim A_j \\to A_i$ satisfy $\\varphi_{ij} \\circ \\pi_j = \\pi_i$.\nFor every algebra $B$ with homomorphisms $\\psi_i \\colon B \\to A_i$ satisfying $\\varphi_{ij} \\circ \\psi_j = \\psi_i$, there is a unique homomorphism $\\psi \\colon B \\to \\varprojlim A_i$ with $\\pi_i \\circ \\psi = \\psi_i$, namely $\\psi(b) = (\\psi_i(b))_i$.\n\nExample: let $p$ be a prime, let $I = \\N$, let $A_n = \\Z/p^n\\Z$ as rings, and let $\\varphi_{mn}$ be reduction modulo $p^m$ for $m \\le n$.\nReduction modulo $p^n$ followed by reduction modulo $p^m$ is reduction modulo $p^m$, so clause 3 holds.\nThe inverse limit is the ring $\\Z_p$ of $p$-adic integers.\n\nNon-example: on $I = \\set{1, 2, 3}$ with the usual order, take $A_i = \\Z/2\\Z$ as groups, $\\varphi_{ii} = \\varphi_{12} = \\varphi_{23} = \\id$ and $\\varphi_{13} = 0$.\nThen $\\varphi_{12} \\circ \\varphi_{23} = \\id \\ne \\varphi_{13}$, so clause 3 fails and the data is not an inverse system.\n\nSome authors allow any partially ordered set, or any small category, as the index; the construction is the same.\n",
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