ON PRIMARY LATTICES
Eizi Inaba · Hokkaido Mathematical Journal · 1948
On Primary Lattices$\Phi$ $\backslash $ with $n=n_{1}+n_{2}$ , since $\frac{I}{e_{1}}\simeq\frac{e_{1}\cdot e_{2}}{e_{1}}'\simeq\frac{e_{2}}{O}$ .Then $C_{i}$ consists of $2^{n_{i}}$ elements by induction-hypothesis. Therefore $C$ consists of $2^{n_{1}}\cdot 2^{n_{2}}=2^{n}-$ elements, and is the direct union of $n$ indecompesable sublattices, since $C_{i}$ is the direc $t$ union of $n_{i}$ indecomposable sublattices.q.e.$d$ .this is not the case, tben we shall say $x$ irreducible.$\cdot$ For example every element from a chain' is irreducible.Every element from a lattice of finite dimension can be represented as the join of irreducible elements.The independent irreducible elements $x_{i}$ with $x=x_{1}\cdot x_{2}\cdots\cdot\cdot x_{ u}$ are called a basis of the element $x$ .If the center $C$ of a modular lattice consists of $2^{n}$ elements, then $I=e_{1}\cdot e_{2}\cdots\cdot\cdot e_{n}$ uniquely with $irreducible$ , elements $e_{i}$ from $C$ and