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  "body": "\n## Definition\n\nLet $\\sigma$ be a signature and let $\\mathbf{A}$ and $\\mathbf{B}$ be $\\sigma$-[[structures|983GZBQYT4]] with domains $A$ and $B$.\nAn *isomorphism* from $\\mathbf{A}$ to $\\mathbf{B}$ is a [[bijection|0GG9TVW90M]] $h \\colon A \\to B$ such that both $h$ and its inverse $h^{-1} \\colon B \\to A$ are [[homomorphisms|KBXSZC6BV1]].\nThe structures $\\mathbf{A}$ and $\\mathbf{B}$ are *isomorphic*, written $\\mathbf{A} \\cong \\mathbf{B}$, if there is an isomorphism from $\\mathbf{A}$ to $\\mathbf{B}$.\n\n## Remarks\n\nThe condition on $h^{-1}$ means that $h$ also reflects relations: $(a_1, \\dots, a_n) \\in R^{\\mathbf{A}}$ if and only if $(h(a_1), \\dots, h(a_n)) \\in R^{\\mathbf{B}}$.\nFor a signature without relation symbols, the inverse of a bijective homomorphism is automatically a homomorphism, so an isomorphism is the same as a bijective homomorphism.\n\nExample: the map $x \\mapsto e^x$ is an isomorphism from $(\\R, +, 0)$ to $(\\R_{\u003e0}, \\cdot, 1)$, because it is a bijection with $e^{x + y} = e^x e^y$ and $e^0 = 1$; its inverse is $\\ln$.\n\nNon-example: let $A = \\set{1, 2}$ with the empty binary relation and $B = \\set{1, 2}$ with the relation $\\le$.\nThe identity map is a bijective homomorphism from $A$ to $B$, but its inverse does not preserve the relation, since $1 \\le 2$ in $B$ while $(1, 2)$ is not related in $A$.\n",
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