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  "body": "\n## Example\n\nLet $V$ be a [[vector space|000AYPA9AJ]] over a field $F$, and let $L(V)$ be the set of all [[linear subspaces|00YC1D652K]] of $V$, ordered by inclusion.\nThe meet of two subspaces $U$ and $W$ is $U \\cap W$, and their join is the sum $U + W = \\set{u + w : u \\in U, w \\in W}$.\nFor any family $(U_i)_{i \\in I}$ of subspaces, the intersection $\\bigcap_i U_i$ and the span of $\\bigcup_i U_i$ are subspaces.\nThey are the greatest lower bound and the least upper bound of the family.\nSo $L(V)$ is a [[complete lattice|D7A0ZGZT5S]], with least element $\\set{0}$ and greatest element $V$.\nIt is a [[modular lattice|QX7WZ92QAK]]: if $U \\subseteq W$, then $U + (X \\cap W) = (U + X) \\cap W$ for every subspace $X$.\nThe inclusion $U + (X \\cap W) \\subseteq (U + X) \\cap W$ holds because $U \\subseteq W$.\nFor the other inclusion, an element $w = u + x$ of the right side with $u \\in U$, $x \\in X$ has $x = w - u \\in W$, so $x \\in X \\cap W$.\n\n## Remarks\n\nThe lattice $L(V)$ is not distributive once $\\dim V \\ge 2$.\nIn $V = \\R^2$, take three distinct lines $a$, $b$, $c$ through $0$.\nThen $a \\cap (b + c) = a \\cap \\R^2 = a$, while $(a \\cap b) + (a \\cap c) = \\set{0}$.\nWhen $V$ is finite-dimensional, the subspaces of dimension $1$ and $2$ are the points and lines of the projective space of $V$, and $L(V)$ encodes its incidence geometry.\n\n## Explanations\n\n### L1 human\n\nTake all the subspaces of a vector space: the zero space, the lines through the origin, the planes through the origin, and so on up to the whole space.\nOrder them by inclusion.\nAny two subspaces have a largest common subspace, their intersection, and a smallest subspace containing both, their sum.\nFor two distinct lines in $\\R^3$, the intersection is $\\set{0}$ and the sum is the plane they span.\nThis ordered set is the basic example of a modular lattice that is not distributive, and it is the algebraic skeleton of projective geometry.\n",
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