Peter Jipsen and Nathan Lawless Dedicated to Brian Davey on the occasion of his 65th birthday
Springer Basel · 2015
Modular lattices, introduced by R. Dedekind, are an important subvariety of lattices that includes all distributive lattices. Heitzig and Reinhold (8) 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. Enumeration of finite mathematical structures is an important tool, since it allows testing new hypotheses and searching for counterexamples. Addi- tionally, it provides insight into the properties of these structures. Here we concentrate on constructing, up to isomorphism, all modular lattices with a given number of elements. The algorithm we develop is a modification of the approach of Heitzig and Reinhold (8) who enumerated (up to isomorphism) all lattices with up to 18 elements. The number of distributive lattices of size up