Lattice (order)
Jordan–Dedekind chain condition

A chain from x 0 {\displaystyle x_{0}} to x n {\displaystyle x_{n}} is a set { x 0, x 1, …, x n }, {\displaystyle \left\{x_{0},x_{1},\ldots,x_{n}\right\},} where x 0 < x 1 < x 2 < … < x n. {\displaystyle x_{0}<x_{1}<x_{2}<\ldots <x_{n}.} The length of this chain is n, or one less than its number of elements. A chain is maximal if x i {\displaystyle x_{i}} covers x i − 1 {\displaystyle x_{i-1}} for all 1 ≤ i ≤ n. {\displaystyle 1\leq i\leq n.}

If for any pair, x {\displaystyle x} and y, {\displaystyle y,} where x < y, {\displaystyle x<y,} all maximal chains from x {\displaystyle x} to y {\displaystyle y} have the same length, then the lattice is said to satisfy the Jordan–Dedekind chain condition.
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