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  "body": "\n## Definition\n\nLet $\\sigma$ be a [[signature|WDVVEYD6AT]] with no function symbols and no constant symbols.\nSo each symbol $R \\in \\sigma$ is a relation symbol with an arity $n_R \\ge 1$.\nA *relational structure* of signature $\\sigma$ is a [[structure|983GZBQYT4]] for $\\sigma$.\nExplicitly, it is a pair $(A, (R^A)_{R \\in \\sigma})$ where $A$ is a nonempty set and $R^A \\subseteq A^{n_R}$ for each $R \\in \\sigma$.\nHere $A^{n_R}$ is the [[Cartesian product|09CEDJYBKP]] of $n_R$ copies of $A$.\n\n## Remarks\n\nExample: a simple graph $(V, E)$, with $E \\subseteq V^2$ the set of pairs $(u, v)$ such that $u$ and $v$ are adjacent, is a relational structure with one binary relation.\n\nNon-example: a group $(G, \\cdot)$ is not a relational structure, because its signature has a function symbol.\nIt can be turned into one by replacing $\\cdot$ with its graph, the ternary relation $\\set{(a, b, c) : a \\cdot b = c}$.\n\nEvery structure can be encoded in this way as a relational structure.\nSome authors allow an empty domain for relational structures.\n",
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