Counting Finite Lattices

Abstract

The correct values for the number of all unlabeled lattices on n elements are known for n ≤ 11. We present a fast orderly algorithm generating all unlabeled lattices up to a given size n. Using this algorithm, we have computed the number of all unlabeled lattices as well as that of all labeled lattices on an n-element set for each n ≤ 18.

Residuated Lattices of Size ≤ 12

“…The first row contains the numbers of non-isomorphic lattices. These numbers agree with observations concerning the numbers of lattices from [20]. The second row contains the numbers of non-isomorphic residuated lattices.…”

“…The problem of counting and listing all non-isomorphic partial orders and, in particular, lattices has been studied in several papers in the past, see e.g. [10,19,20,24,25,28,29], see also Chapter XI in [12]. In [20], the numbers of all finite lattices with up to 18 elements are presented along with the algorithm for listing the lattices.…”

“…[10,19,20,24,25,28,29], see also Chapter XI in [12]. In [20], the numbers of all finite lattices with up to 18 elements are presented along with the algorithm for listing the lattices. For our purposes, we could have used the algorithm from [20].…”

“…In [20], the numbers of all finite lattices with up to 18 elements are presented along with the algorithm for listing the lattices. For our purposes, we could have used the algorithm from [20]. However, we briefly describe another algorithm, which we used for computing finite lattices.…”

Counting graded lattices of rank 3 that have few coatoms

“…Previous works that count lattices fall roughly into two categories. For small lattices, exact counts are often determined by constructive methods, by actually generating all lattices up to a certain size [8,10,12,16], or lattices in some specific family [6,12]. For large lattices there are asymptotic upper and lower bounds [6,12,13,14].…”