An inductive algorithm to construct finite lattices

Abstract:

G. Birkhoff [1] proposed the following problem: Enumerate all finite lattices which are uniquely determined (up to isomorphism) by their diagram, considered as a graph. It is not known how many lattices of order n exist, except when the value of n is quite small. The aim of this note is to give an algorithm to construct inductively all finite lattices of order n. Using this algorithm, we have found that there exist 222 lattices for $n = 8$ and 1078 lattices for $n = 9$. All lattices of order $n \leqslant 8$ are shown at the end of this note.