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  "body": "\n## Definition\n\nLet $A$ be an algebra of a signature $\\sigma$ with only function symbols, and let $n \\ge 1$.\nLet $t$ be a [[term|6D3WA7N2W4]] over $\\sigma$ whose variables are among $x_1, \\ldots, x_n$.\nThe $n$-ary *term operation* of $t$ on $A$ is the map $t^A \\colon A^n \\to A$ given by\n$$t^A(a_1, \\ldots, a_n) = t^A[\\alpha].$$\nHere $\\alpha(x_i) = a_i$, and $t^A[\\alpha]$ is the [[value|DH30WTA26T]] of $t$ at the assignment $\\alpha$.\nAn operation on $A$ is a *term operation* of $A$ if it equals $t^A$ for some $n$ and some term $t$.\n\n## Remarks\n\nThe term operation of the variable $x_i$ is the projection $(a_1, \\ldots, a_n) \\mapsto a_i$.\nThe term operation of $f(t_1, \\ldots, t_k)$ is the composition $f^A(t_1^A, \\ldots, t_k^A)$.\nSo the set $\\operatorname{Clo}(A)$ of all term operations is the smallest set of operations on $A$ that contains the projections and the basic operations and is closed under composition.\nEvery homomorphism $h \\colon A \\to B$ satisfies $h(t^A(a_1, \\ldots, a_n)) = t^B(h(a_1), \\ldots, h(a_n))$.\n\nExample: in the group $(\\Z, +, -, 0)$ the map $(a, b) \\mapsto 2a - b$ is the term operation of $(x_1 + x_1) + (-x_2)$.\n\nNon-example: the map $a \\mapsto a + 1$ on $(\\Z, +, -, 0)$ is not a term operation.\nEvery term operation $t^A$ of this group satisfies $t^A(0, \\ldots, 0) = 0$, since $0 + 0 = -0 = 0$, while $0 + 1 \\ne 0$.\n",
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