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  "body": "\n## Definition\n\nLet $(P, L, I)$ be an [[incidence structure|FGKNAFNHA3]] with point set $P$, line set $L$ and incidence relation $I \\subseteq P \\times L$.\nIt is a *partial linear space* if:\n\n1. any two distinct points are incident with at most one common line;\n2. every line is incident with at least two points.\n\nIt is a *linear space* if in clause 1 \"at most one\" is replaced by \"exactly one\".\n\n## Remarks\n\nExample: the $3 \\times 3$ grid, with $P = \\set{1, 2, 3}^2$ and $L$ the three rows $\\set{i} \\times \\set{1, 2, 3}$ and the three columns $\\set{1, 2, 3} \\times \\set{j}$.\nA row and a column meet in one point, two rows or two columns are disjoint, so two distinct points share at most one line; each line has three points.\n\nNon-example: $P = \\set{a, b, c}$ with lines $\\set{a, b}$ and $\\set{a, b, c}$.\nThe points $a$ and $b$ lie on two lines, so clause 1 fails.\n\nClause 1 is equivalent to its dual: two distinct lines meet in at most one point.\nPartial linear spaces are also called semilinear spaces or near-linear spaces.\n",
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