It has been unknown whether Hirota's discrete Korteweg-de Vries equation and the lattice sine-Gordon have consistency around a cube (CAC) property. In this paper, we show that they CAC Moreover, also can be extended to systems on 3-dimensional integer lattice.

The problem of maximizing non-negative submodular functions has been studied extensively in the last few years. However, most papers consider submodular set functions. Recently, several advances have been made for submodular functions on the integer lattice. As a direct generalization of submodular set functions, a function f : {0, . . . , C}n → R+ is submodular, if f(x)+ f(y) ≥ f(x∧ y)+ f(x∨ y...

Every lattice is the complete join of all its one-element sublattices. In this paper we address the question: Which lattices L have the property that L is finitely join reducible in SubL? That is, when do there exist proper sublattices A, B such that L = A∨B? In particular, could it be that every nontrivial lattice has this property, in which case every element of SubL would be finitely join re...

Various embedding problems of lattices into complete lattices are solved. We prove that for any join-semilattice S with the minimal join-cover refinement property, the ideal lattice IdS of S is both algebraic and dually algebraic. Furthermore, if there are no infinite D-sequences in J(S), then IdS can be embedded into a direct product of finite lower bounded lattices. We also find a system of i...

In Part I of [3], Adaricheva and Nation introduced a new property that holds (dually) for the natural equaclosure operator on congruence lattices of semilattices with operators, and hence holds in lattices of quasivarieties Lq(K). The new property implied some previously known conditions as well. Using the duality for congruences of semilattices with operators from [7], we will refine that to a...

Digital snowflakes are solidifying cellular automata on the triangular lattice with the property that a site having exactly one occupied neighbor always becomes occupied at the next time. We demonstrate that each such rule fills the lattice with an asymptotic density that is independent of the initial finite set. There are some cases in which this density can be computed exactly, and others in ...