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  "body": "\n## Statement\n\nLet $L$ be a [[modular lattice|QX7WZ92QAK]] and let $a \\le b$ in $L$.\nWrite $x \\lessdot y$ when $y$ [[covers|4HHDPZ4BDY]] $x$.\nFor $u \\le v$ in $L$, write $[u, v] = \\set{z \\in L : u \\le z \\le v}$.\nTwo such intervals are *transposed* if they have the form $[x \\wedge y, x]$ and $[y, x \\vee y]$ for some $x, y \\in L$, in either order.\nTwo intervals are *projective* if they are joined by a finite sequence of intervals in which each consecutive pair is transposed.\n\nLet\n$$a = x_0 \\lessdot x_1 \\lessdot \\cdots \\lessdot x_m = b \\quad\\text{and}\\quad a = y_0 \\lessdot y_1 \\lessdot \\cdots \\lessdot y_n = b$$\nbe two finite [[maximal chains|3AX549FFFM]] from $a$ to $b$.\nThen $m = n$, and there is a permutation $\\sigma$ of $\\set{1, \\dots, m}$ such that $[x_{i-1}, x_i]$ is projective to $[y_{\\sigma(i)-1}, y_{\\sigma(i)}]$ for each $i$.\n\n## Remarks\n\nThe proof is by induction on $m$.\nIf $x_1 = y_1$, apply induction to the interval $[x_1, b]$.\nOtherwise $x_1$ and $y_1$ both cover $a = x_1 \\wedge y_1$, and modularity makes the transposed intervals $[y_1, x_1 \\vee y_1]$ and $[a, x_1]$ isomorphic, so $x_1 \\vee y_1$ covers both $x_1$ and $y_1$.\nChoosing a maximal chain from $x_1 \\vee y_1$ to $b$ and applying induction in $[x_1, b]$ and in $[y_1, b]$ gives the result.\n\nExample: the lattice of submodules of a module $M$ is modular.\nFor submodules $X$ and $Y$, the transposed intervals give the isomorphism $X / (X \\cap Y) \\cong (X + Y) / Y$, so projective intervals of length one give isomorphic simple quotients.\nSo the theorem yields the [[Jordan–Hölder theorem for modules|M1ABX18FC4]]: two [[composition series|0XV2A49X6Q]] of a module have the same length and isomorphic factors up to order.\n\nWithout modularity the equal-length conclusion can fail.\nIn the [[pentagon lattice N5|DHC8TZXGYS]], the maximal chains $0 \u003c a \u003c c \u003c 1$ and $0 \u003c b \u003c 1$ have lengths $3$ and $2$.\n\n## Explanations\n\n### L1 human\n\nIn a modular lattice, any two ways of climbing from $a$ to $b$ by smallest possible steps use the same number of steps.\nMoreover, the steps of one climb match the steps of the other, up to order, in a precise sense.\nThis is the lattice form of the familiar fact that composition series of a module have the same length and the same factors.\n\n### L2 human\n\nThe theorem shows that modular lattices satisfy the [[Jordan–Dedekind chain condition|RW7YC3HG0S]] on every interval of finite length, so such intervals have a well-defined length.\nFor submodule lattices it recovers the Jordan–Hölder theorem for modules, with projectivity of intervals giving isomorphism of factors.\nThe [[Jordan–Hölder theorem|0E088ZP533]] for groups is a close relative, with subnormal series in place of chains of submodules.\nThe same equal-length conclusion holds in [[semimodular lattices|C1NET23GG0]], which is why matroids and geometric lattices have a rank function.\n",
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