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  "body": "\n## Problem\n\nLet $\\mathcal{V}$ be a [[variety|AJNZJD8442]] of finite signature with a finite [[equational basis|9Q008GMZSW]] $\\Sigma$.\nLet $X$ be a countably infinite set of variables, and let $F_{\\mathcal{V}}(X)$ be the [[free algebra|GTS4DRRMZR]] of $\\mathcal{V}$ on $X$.\nThe problem 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 $F_{\\mathcal{V}}(X)$.\nEquivalently, it asks whether the [[equational theory|F1M51B4MDG]] of $\\mathcal{V}$ is decidable.\nThat is, can one decide whether the [[identity|PKD440G2XW]] $s \\approx t$ holds in every algebra of $\\mathcal{V}$?\n\n## Remarks\n\nSince $\\Sigma$ is finite, the identities of $\\mathcal{V}$ are computably enumerable by listing all derivations in equational logic; the difficulty is to certify that an identity fails.\nThe problem is solvable for many classical varieties.\nIn groups, every element of a free group has a unique reduced word.\nIn semigroups and monoids, the free algebras are the algebras of words, so equality is literal equality of words.\nIn Boolean algebras, an identity holds exactly when it holds in the two-element Boolean algebra, which can be checked by truth tables.\nIt is unsolvable in general: Tarski showed that the equational theory of relation algebras, which form a finitely based variety, is undecidable.\n\n## Explanations\n\n### L1 human\n\nFix a list of laws, such as the group laws.\nThe question is whether there is a mechanical test that decides if an equation between two expressions follows from those laws.\nFor groups the test is to cancel $x x^{-1}$ until nothing cancels and compare the results.\nFor some finite lists of laws, no such test exists.\n",
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      "first_stated": "1914",
      "notes": "Thue posed the word problem for finitely presented semigroups in 1914, after Max Dehn posed it for groups in 1911.",
      "posed_by": "Axel Thue"
    },
    "kind": "problem",
    "msc": [
      "08A50",
      "08B20",
      "03B25"
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    "name": "Word problem for free algebras of a variety",
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        "url": "https://en.wikipedia.org/wiki/Word_problem_(mathematics)"
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  "link": "https://tamnd.github.io/math2/node/AR14VMVHNG/",
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  "msc": [
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  "name": "Word problem for free algebras of a variety",
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  "source": "https://github.com/tamnd/math2/blob/main/nodes/AR1/AR14VMVHNG.md",
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