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  "body": "\n## Definition\n\nLet $(I, \\le)$ be a nonempty [[directed set|D9RY4667K6]].\nA *directed system* of algebras over $I$ consists of [[algebras|QJGPT2W0FT]] $\\mathbf{A}_i$ of one signature, for $i \\in I$, and [[homomorphisms|KBXSZC6BV1]] $f_{ij} \\colon \\mathbf{A}_i \\to \\mathbf{A}_j$ for $i \\le j$, such that:\n\n1. $f_{ii}$ is the identity of $\\mathbf{A}_i$ for each $i$;\n2. $f_{jk} \\circ f_{ij} = f_{ik}$ whenever $i \\le j \\le k$.\n\nOn the disjoint union of the universes, let $a \\sim b$ for $a \\in A_i$ and $b \\in A_j$ when $f_{ik}(a) = f_{jk}(b)$ for some $k \\ge i, j$.\nThe *direct limit* $\\varinjlim \\mathbf{A}_i$ has as universe the set of classes $[a]$ of this [[equivalence relation|0G8VEZ3WTK]].\nFor an $n$-ary operation $f$ and $a_m \\in A_{i_m}$, choose $k \\ge i_1, \\dots, i_n$ and put $f([a_1], \\dots, [a_n]) = [f^{\\mathbf{A}_k}(f_{i_1 k}(a_1), \\dots, f_{i_n k}(a_n))]$.\n\n## Remarks\n\nThe relation $\\sim$ is transitive and the operations do not depend on the choices of $k$ and representatives, because $I$ is directed and the $f_{ij}$ are homomorphisms.\nFor a nullary operation $c$, take any $k \\in I$ and put $c = [c^{\\mathbf{A}_k}]$.\nThe maps $a \\mapsto [a]$ are homomorphisms $\\mathbf{A}_i \\to \\varinjlim \\mathbf{A}_i$, and they make the direct limit a colimit of the system.\n\nExample: let $\\mathbf{A}_n = \\Z/p^n\\Z$ for $n \\ge 1$, with $f_{nm}(x) = p^{m-n} x$ for $n \\le m$.\nThese maps are injective homomorphisms satisfying clauses 1 and 2, and the direct limit is the Prüfer $p$-group.\n\nNon-example: over $I = \\N$, take $\\mathbf{A}_n = \\Z$ for all $n$ with $f_{nm}(x) = 2x$ for all $n \u003c m$ and $f_{nn} = \\id$.\nClause 2 fails, since $f_{12} \\circ f_{01}(x) = 4x$ while $f_{02}(x) = 2x$.\nSetting instead $f_{nm}(x) = 2^{m-n} x$ gives a directed system with direct limit $\\Z[1/2]$.\n",
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