Canonical Subdirect Factorizations of Lattices
Toshio Tanaka · Hiroshima Mathematical Journal · 1952
G. Birkhoff has proved the following factorization theoremn:Every algebra A with finitary operations can be represented as a subdirect union of subdirectly irreducible algebras.And the subdirect factorizations are closely related to the structure of the lattice 2 ) <~(A) of all congruence relations on A. In particular, for a lattice L <8'J(L) is pseudo-complemented.Using this fact F. Maeda [1] has introduced the canonical subdirect factorization of lattices in which the unique factorization theorem is proved.In this paper, I show that a lattice L has the canonical subdirect factorization with subdirectly irreducible factors if and only if c~)(L) is an atomic lattice.And also I give a solution for the Birkhoff's problem 72, 3 ) that is, a necessary and sufficient condition such that ®(L) is a Boolean algebra is that L has a subdirect factorization with simple factors such that the components of arbitrary two elements of L ar.e identical except