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  "body": "\n## Definition\n\nLet $\\Sigma$ be a signature, that is, a set of function symbols, each with an [[arity|MSXX2P86CY]] $n \\ge 0$.\nA *partial $\\Sigma$-algebra* is a [[set|0SJ3RX9TXA]] $A$ with the following data for each symbol $f \\in \\Sigma$ of arity $n$.\nIt has a subset $\\operatorname{dom}(f^A)$ of the [[Cartesian power|09CEDJYBKP]] $A^n$ and a map $f^A \\colon \\operatorname{dom}(f^A) \\to A$.\nThe map $f^A$ is a *partial operation* on $A$, and $f^A(a_1, \\dots, a_n)$ is defined exactly when $(a_1, \\dots, a_n) \\in \\operatorname{dom}(f^A)$.\n\n## Remarks\n\nWhen $\\operatorname{dom}(f^A) = A^n$ for every $f$, the partial algebra is an ordinary (total) algebra.\n\nExample: the natural numbers $\\N$ with subtraction, where $m - n$ is defined exactly when $m \\ge n$.\nThe domain is $\\set{(m, n) \\in \\N^2 : m \\ge n}$, and on it $m - n \\in \\N$, so subtraction is a partial operation on $\\N$.\nIt is not total, since $(0, 1)$ is not in its domain.\n\nNon-example: subtraction on all of $\\N^2$ with values in $\\Z$ is not a partial operation on $\\N$, since its value $0 - 1 = -1$ does not lie in the carrier $\\N$.\n\nA field $K$ with the partial operation $x \\mapsto x^{-1}$ defined on $K \\setminus \\set{0}$ is a partial algebra for the signature $\\set{+, \\cdot, -, ^{-1}, 0, 1}$.\nHomomorphisms, subalgebras and congruences admit several inequivalent versions for partial algebras, for example weak and strong homomorphisms, so the version in use must be stated.\n",
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