A lattice $L$ is said to satisfy (the lattice theoretic version of) Frankl’s conjecture if there is a join-irreducible element $f\in L$ such that at most half of the elements $x$ of $L$ satisfy $f\le x$. Frankl’s conjecture, also called as union-closed sets conjecture, is well-known in combinatorics, and it is equivalent to the statement that every finite lattice satisfies Frankl’s conjecture. Let $m$ denote the number of nonzero join-irreducible elements of $L$. It is well-known that $L$ consists of at most $2^m$ elements....