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  "body": "\n## Definition\n\nLet $S$ be a [[set|0SJ3RX9TXA]], whose elements are called *sorts*.\nAn *$S$-sorted signature* $\\Sigma$ is a set of operation symbols, each with a type $s_1 \\cdots s_n \\to s$, where $n \\ge 0$ and $s_1, \\dots, s_n, s \\in S$.\nA *many-sorted $\\Sigma$-algebra* $A$ consists of:\n\n1. a set $A_s$ for each sort $s \\in S$, the carrier of sort $s$;\n2. for each symbol $f$ of type $s_1 \\cdots s_n \\to s$, a [[function|0EPAT51DXQ]] $f^A \\colon A_{s_1} \\times \\cdots \\times A_{s_n} \\to A_s$ on the [[Cartesian product|09CEDJYBKP]].\n\nA *homomorphism* $h \\colon A \\to B$ of $\\Sigma$-algebras is a family of maps $h_s \\colon A_s \\to B_s$ with $h_s(f^A(a_1, \\dots, a_n)) = f^B(h_{s_1}(a_1), \\dots, h_{s_n}(a_n))$ for every symbol $f$ of type $s_1 \\cdots s_n \\to s$ and all $a_i \\in A_{s_i}$.\n\n## Remarks\n\nExample: let $S = \\set{\\mathrm{scalar}, \\mathrm{vector}}$, with symbols $+ \\colon \\mathrm{vector}\\,\\mathrm{vector} \\to \\mathrm{vector}$, $\\cdot \\colon \\mathrm{scalar}\\,\\mathrm{vector} \\to \\mathrm{vector}$, and the field operations on the sort $\\mathrm{scalar}$.\nA vector space $V$ over a field $K$ gives an algebra with $A_{\\mathrm{scalar}} = K$ and $A_{\\mathrm{vector}} = V$; each operation sends arguments of the right sorts to the right sort.\n\nNon-example: the same data with a map $K \\times V \\to K$ in place of scalar multiplication fails clause 2, since the value of $\\cdot$ must lie in the carrier of sort $\\mathrm{vector}$.\n\nWith one sort, a many-sorted algebra is an ordinary algebra.\nBirkhoff and Lipson called these structures *heterogeneous algebras*.\nSome authors require each carrier $A_s$ to be nonempty.\n",
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