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  "body": "\n## Problem\n\nLet $\\mathcal{V}$ be a [[variety|AJNZJD8442]] of finite signature, and let $A = \\langle X \\mid R \\rangle_{\\mathcal{V}}$ be an algebra given by a finite [[presentation|43M1WFPYGQ]], with $X$ and $R$ finite.\nThe *word problem* for this presentation asks for an algorithm, that is, a [[computable function|KZSB5E471K]], with the following task.\nFor any two terms $s$ and $t$ over $X$, it decides whether $s$ and $t$ denote the same element of $A$.\nThe problem is *solvable* for $A$ if such an algorithm exists.\nThe *uniform word problem* for $\\mathcal{V}$ asks for one algorithm that takes the finite presentation as part of its input.\n\n## Remarks\n\nSolvability does not depend on the choice of the finite presentation of $A$.\nSuppose that $\\mathcal{V}$ is finitely based.\nThen the set of pairs $(s, t)$ with $s = t$ in $A$ is computably enumerable, by listing all derivations from $R$ and the identities of $\\mathcal{V}$.\nThe difficulty is to certify that $s \\neq t$.\nThe problem is unsolvable in general.\nMarkov and Post independently gave finitely presented semigroups with unsolvable word problem in 1947.\nNovikov and Boone gave finitely presented groups with unsolvable [[word problem for groups|PBGTTAVMMT]].\nIt is solvable in many classes, for example for finitely presented abelian groups and for one-relator groups, the latter by a theorem of Magnus.\nWhen the relations together with the identities of $\\mathcal{V}$ can be oriented into a finite convergent rewriting system, normal forms solve it.\n\n## Explanations\n\n### L1 human\n\nAn algebra given by generators and relations is like a puzzle: different expressions may name the same element because the relations say so.\nThe word problem asks for a mechanical procedure that always tells whether two given expressions are equal.\nFor many familiar algebras such a procedure exists.\nA famous discovery of the twentieth century is that for some finitely presented semigroups and groups no procedure can exist at all.\n",
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    "name": "Word problem for an algebra",
    "sources": [
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        "url": "https://en.wikipedia.org/wiki/Word_problem_(mathematics)"
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  "markdown": "https://tamnd.github.io/math2/node/7ZKBV18BFG.md",
  "msc": [
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  "name": "Word problem for an algebra",
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