Order, lattices, ordered algebraic structures Modular and complemented lattices Theorem

Jordan–Hölder theorem for modular lattices

Builds on Modular lattice Covering relation Maximal chain

Statement

Let $L$ be a modular lattice and let $a \le b$ in $L$ . Write $x \lessdot y$ when $y$ covers $x$ . For $u \le v$ in $L$ , write $[u, v] = \set{z \in L : u \le z \le v}$ . Two 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. Two intervals are projective if they are joined by a finite sequence of intervals in which each consecutive pair is transposed.

Let $a = x_0 \lessdot x_1 \lessdot \cdots \lessdot x_m = b \quad\text{and}\quad a = y_0 \lessdot y_1 \lessdot \cdots \lessdot y_n = b$ be two finite maximal chains from $a$ to $b$ . Then $m = n$ , 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 $i$ .

Remarks

The proof is by induction on $m$ . If $x_1 = y_1$ , apply induction to the interval $[x_1, b]$ . Otherwise $x_1$ 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 $y_1$ . Choosing 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.

Example: the lattice of submodules of a module $M$ is modular. For 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. So the theorem yields the Jordan–Hölder theorem for modules: two composition series of a module have the same length and isomorphic factors up to order.

Without modularity the equal-length conclusion can fail. In the pentagon lattice N5, the maximal chains $0 < a < c < 1$ and $0 < b < 1$ have lengths $3$ and $2$ .

Going deeper

For a student

The theorem shows that modular lattices satisfy the Jordan–Dedekind chain condition on every interval of finite length, so such intervals have a well-defined length. For submodule lattices it recovers the Jordan–Hölder theorem for modules, with projectivity of intervals giving isomorphism of factors. The Jordan–Hölder theorem for groups is a close relative, with subnormal series in place of chains of submodules. The same equal-length conclusion holds in semimodular lattices, which is why matroids and geometric lattices have a rank function.

Reading path

To understand this page from the ground up, read these 4 pages in order. The foundations come first.

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Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.

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Covering relation concept
Maximal chain concept
Modular lattice concept
Jordan–Hölder theorem for modular lattices statement

Tag	`BCG5KM83AP`
URI	`m2://BCG5KM83AP`
Kind	statement, theorem
Writing	W0 stub
MSC	06C05, 06B05
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About	Modular lattice
Uses	Modular lattice, Covering relation, Maximal chain