Direct decompositions of atomistic algebraic lattices
A lattice is atomistic if each of its elements is a join of atoms. An element \(x\) of a complete lattice \(L\) is strictly join-irreducible if \(x= \bigvee X\) implies \(x\in X\) for any subset \(X\subseteq L\). A lattice \(L\) is called a \(V_ 1\)-lattice if each its elements is the join of strictly join-irreducible elements of \(L\). Main results: Theorem 1. Every atomistic algebraic lattice is a direct product of directly indecomposable (atomistic algebraic) lattices. Theorem 2. Every algebraic \(V_ 1\)-lattice is a direct product of directly indecomposable (algebraic) \(V_ 1\)-lattices.
scientific article; zbMATH DE number 724106