A note on the number of irreducibles of subdirect products (Q1088691)

Let J(L) (resp. M(L)) be the set of nonzero (nonunit) join (meet) irreducible elements of a finite lattice L and Con(L) its congruence lattice. If L is modular and a subdirect product of subdirectly irreducible lattices \(L_ 1,...,L_ n\) then \(| J(L)| =\sum_{i}| J(L_ i)|\). The author proves this as a corollary to his more general theorem: The conditions (a) \(| J(L)| =\sum_{i}| J(L_ i)|\), (b) \(| M(L)| =\sum_{i}| M(L_ i)|\), (c) Con(L)\(\cong \prod_{i}Con(L_ i)\) are equivalent, where L is the subdirect product of lattices \(L_ 1,...,L_ n\).