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  "body": "\n## Statement\n\nLet $\\mathcal{V}$ be a [[variety|AJNZJD8442]] of finite signature.\nA finite [[partial algebra|H7RX5729ZB]] $P$ of the same signature *embeds in* $\\mathcal{V}$ if there are $B \\in \\mathcal{V}$ and an injective map $e \\colon P \\to B$ with the following property.\nWhenever $f(p_1, \\ldots, p_n)$ is defined in $P$, one has $e(f(p_1, \\ldots, p_n)) = f(e(p_1), \\ldots, e(p_n))$ in $B$.\nThen the following are equivalent:\n\n1. The uniform [[word problem|7ZKBV18BFG]] for finitely presented algebras of $\\mathcal{V}$ is solvable.\n2. There is an algorithm that decides, for any finite partial algebra $P$ of the signature, whether $P$ embeds in $\\mathcal{V}$.\n\n## Remarks\n\nFrom 1 to 2: take the [[presentation|43M1WFPYGQ]] whose generators are the elements of $P$ and whose relations are the entries of the partial operation tables.\nThen $P$ embeds in $\\mathcal{V}$ exactly when distinct elements of $P$ stay distinct in the presented algebra.\nFrom 2 to 1: given relations $R$ and terms $s, t$, let $S$ be the finite set of subterms of $s$, $t$ and the terms in $R$.\nThen $s \\ne t$ in $\\langle X \\mid R \\rangle_{\\mathcal{V}}$ exactly when some quotient of $S$, as a partial algebra, satisfies $R$, separates $s$ and $t$, and embeds in $\\mathcal{V}$.\nThere are finitely many quotients to test.\nA useful sufficient condition: if $\\mathcal{V}$ is finitely based and every finite partial algebra that embeds in $\\mathcal{V}$ embeds in a finite algebra of $\\mathcal{V}$, then condition 2 holds.\nIndeed, one can search at the same time for two things.\nOne is a finite algebra of $\\mathcal{V}$ in which $P$ embeds.\nThe other is an equational derivation that identifies two distinct elements of $P$ in the presented algebra.\nExample: Boolean algebras are finitely based.\nA finite partial Boolean algebra that embeds in some Boolean algebra embeds in the finite subalgebra it generates.\nSo finitely presented Boolean algebras have solvable word problem.\n\n## Explanations\n\n### L1 human\n\nThe word problem asks whether two expressions are equal in an algebra given by generators and relations.\nThe theorem says that this is the same question as deciding whether a small finite table of partial operations can be completed inside some algebra of the variety.\nSo word problems can be attacked by thinking about finite pieces of algebras.\n\n### L2 human\n\nThe theorem, due to Trevor Evans, connects the word problem with embeddings of partial algebras.\nCombined with the finite embeddability property, it explains why finitely based varieties such as Boolean algebras have solvable word problems.\nIt also explains why unsolvability results for semigroups and groups are hard: one must build finite partial algebras whose embeddability encodes a halting problem.\n",
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