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  "body": "\n## Definition\n\nLet $\\Sigma$ be a signature of function symbols and let $K$ be a class of $\\Sigma$-algebras.\nAn algebra $I \\in K$ is *initial in $K$* if for every $A \\in K$ there is exactly one [[homomorphism|KBXSZC6BV1]] $I \\to A$.\nThe [[term algebra|3YYKGWJJND]] $T_\\Sigma$ of ground terms is initial in the class of all $\\Sigma$-algebras.\nIn *initial algebra semantics*, the meaning of a ground term $t$ in a $\\Sigma$-algebra $A$ is $[\\![t]\\!]_A = h_A(t)$, where $h_A \\colon T_\\Sigma \\to A$ is the unique homomorphism.\nLet $E$ be a set of [[equations|PKD440G2XW]].\nThe meaning of the specification $(\\Sigma, E)$ is the initial algebra of the class of $\\Sigma$-algebras that satisfy $E$.\nIt is determined up to isomorphism.\n\n## Remarks\n\nSince $h_A$ is a homomorphism, $[\\![f(t_1, \\dots, t_n)]\\!]_A = f^A([\\![t_1]\\!]_A, \\dots, [\\![t_n]\\!]_A)$, so meanings are compositional.\nThe initial algebra of the algebras satisfying $E$ exists: it is the quotient of $T_\\Sigma$ by the least congruence that contains all ground instances of the equations of $E$.\n\nExample: let $\\Sigma = \\set{0, s, +}$ with arities $0, 1, 2$, and let $E$ consist of $x + 0 = x$ and $x + s(y) = s(x + y)$.\nThe natural numbers $\\N$ with zero, successor and addition satisfy $E$.\nEvery ground term equals some $s^n(0)$ by $E$, and distinct numerals $s^m(0) \\ne s^n(0)$ are separated in $\\N$, so $\\N$ is initial among the models of $E$.\n\nNon-example: $\\N$ is not initial in the class of all $\\Sigma$-algebras.\nIn $T_\\Sigma$ the terms $s(0)$ and $s(0) + 0$ are different, but in $\\N$ they have the same value $1$.\nA homomorphism $\\N \\to T_\\Sigma$ would send $1$ to both terms, so there is none.\n\nThe general notion of an [[initial algebra|R76EC56V9R]] for a functor contains this one: $T_\\Sigma$ is the initial algebra of the polynomial functor of $\\Sigma$.\n",
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