Cartesian Lattice Counting by the Vertical 2-sum

A vertical 2-sum of a two-coatom lattice L and a two-atom lattice U is obtained by removing the top of L and the bottom of U, and identifying the coatoms of L with the atoms of U. This operation creates one or two nonisomorphic lattices depending on the symmetry case. Here the symmetry cases are analyzed, and a recurrence relation is presented that expresses the number of nonisomorphic vertical 2-sums in some desired family of graded lattices. Nonisomorphic, vertically indecomposable modular and distributive lattices are counted and classified up to 35 and 60 elements respectively. Asymptotically their numbers are shown to be at least Ω(2.3122n) and Ω(1.7250n), where n is the number of elements. The number of semimodular lattices is shown to grow faster than any exponential in n.

Medians are Below Joins in Semimodular Lattices of Breadth 2

Let L be a lattice of finite length and let d denote the minimum path length metric on the covering graph of L. For any $\xi =(x_{1},\dots ,x_{k})\in L^{k}$ ξ = ( x 1 , … , x k ) ∈ L k , an element y belonging to L is called a median of ξ if the sum d(y,x1) + ⋯ + d(y,xk) is minimal. The lattice L satisfies the c1-median property if, for any $\xi =(x_{1},\dots ,x_{k})\in L^{k}$ ξ = ( x 1 , … , x k ) ∈ L k and for any median y of ξ, $y\leq x_{1}\vee \dots \vee x_{k}$ y ≤ x 1 ∨ ⋯ ∨ x k . Our main theorem asserts that if L is an upper semimodular lattice of finite length and the breadth of L is less than or equal to 2, then L satisfies the c1-median property. Also, we give a construction that yields semimodular lattices, and we use a particular case of this construction to prove that our theorem is sharp in the sense that 2 cannot be replaced by 3.

An extension of F. Šik’s theorem on modular lattices

J. Jakubík noted in [JAKUBÍK, J.:Modular Lattice of Locally Finite Length, Acta Sci. Math.37(1975), 79–82] that F. Šik in the unpublished manuscript proved that in the class of upper semimodular lattices of locally finite length, modularity is equivalent to the lack of cover-preserving sublattices isomorphic toS7. In the present paper we extend the scope of Šik’s theorem to the class of upper semimodular, upper continuous and strongly atomic lattices. Moreover, we show that corresponding result of Jakubík from [JAKUBÍK, J.:Modular Lattice of Locally Finite Length, Acta Sci. Math.37(1975), 79–82] cannot be strengthened is analogous way.