Paper detail

Generating all finite modular lattices of a given size

Modular lattices, introduced by R. Dedekind, are an important subvariety of lattices that includes all distributive lattices. Heitzig and Reinhold developed an algorithm to enumerate, up to isomorphism, all finite lattices up to size 18. Here we adapt and improve this algorithm to construct and count modular lattices up to size 24, semimodular lattices up to size 22, and lattices of size 19. We also show that $2^{n-3}$ is a lower bound for the number of nonisomorphic modular lattices of size $n$.