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  "body": "\n## Definition\n\nLet $S$ be a set of sorts and let $\\Sigma$ be an $S$-sorted signature.\nAn *algebraic specification* is a pair $(\\Sigma, E)$, where $E$ is a set of [[equations|PKD440G2XW]] between $\\Sigma$-terms with variables, each side of the same sort.\nA *model* of $(\\Sigma, E)$ is a [[many-sorted $\\Sigma$-algebra|WAQJJ5GX68]] that satisfies every equation of $E$.\nA model $I$ is *initial* if for every model $A$ there is exactly one [[homomorphism|KBXSZC6BV1]] $I \\to A$.\nThe *abstract data type* specified by $(\\Sigma, E)$ is the class of initial models of $(\\Sigma, E)$.\n\n## Remarks\n\nInitial models exist, and any two of them are isomorphic, so the abstract data type is one isomorphism class of algebras; this is the [[initial algebra semantics|HVF403PQ4H]] of the specification.\nAn initial model has two features.\nEvery element is the value of a ground term (\"no junk\"), and two ground terms are equal in it only if $E$ proves them equal (\"no confusion\").\n\nExample: sorts $\\mathrm{nat}$ and $\\mathrm{bool}$; constants $\\mathrm{true}, \\mathrm{false} \\colon \\mathrm{bool}$ and $0 \\colon \\mathrm{nat}$; operations $s \\colon \\mathrm{nat} \\to \\mathrm{nat}$ and $\\mathrm{iszero} \\colon \\mathrm{nat} \\to \\mathrm{bool}$; equations $\\mathrm{iszero}(0) = \\mathrm{true}$ and $\\mathrm{iszero}(s(x)) = \\mathrm{false}$.\nThe model with carriers $\\N$ and $\\set{\\mathrm{true}, \\mathrm{false}}$ is initial.\nEach element is the value of a ground term, and two ground terms with the same value are equal by the equations.\n\nNon-example: the model with one element in each sort satisfies $E$ but is not initial.\nA homomorphism from it to the model above would send its single Boolean element to both $\\mathrm{true}$ and $\\mathrm{false}$, which is impossible.\n",
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