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  "body": "\n## Definition\n\nAn *incidence structure* is a triple $(P, L, I)$ in which $P$ and $L$ are disjoint sets and $I \\subseteq P \\times L$ is a [[binary relation|05AVARPXA2]] from $P$ to $L$.\nThe elements of $P$ are the *points*, the elements of $L$ are the *lines* (or *blocks*), and $I$ is the *incidence relation*.\nA point $p$ and a line $\\ell$ are *incident*, and one says that $p$ lies on $\\ell$, when $(p, \\ell) \\in I$.\n\n## Remarks\n\nExample: let $P = \\set{1, 2, 3}$, let $L = \\set{a, b}$, and let $I = \\set{(1, a), (2, a), (2, b), (3, b)}$.\nCheck: $P$ and $L$ are disjoint and $I$ is a subset of $P \\times L$, so $(P, L, I)$ is an incidence structure, a path of two lines meeting at the point $2$.\n\nNon-example: with the same $P$ and $L$, a set $I$ that contains the pair $(1, 2)$ is not an incidence relation, because $(1, 2)$ is not in $P \\times L$.\n\nThe *dual* of $(P, L, I)$ is $(L, P, I^{\\top})$, where $(\\ell, p) \\in I^{\\top}$ exactly when $(p, \\ell) \\in I$.\nAn *isomorphism* from $(P, L, I)$ to $(P', L', I')$ is a pair of bijections $P \\to P'$ and $L \\to L'$ that preserves incidence in both directions.\nOften lines are taken to be subsets of $P$, with incidence given by membership.\nThe general definition also allows two different lines with the same set of points.\nA finite incidence structure is described by its [[incidence matrix|9DNFN8M83X]].\n",
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