A completion for distributive nearlattices
The aim of this article is to propose an adequate completion for distributive nearlattices. We give a proof of the existence of such a completion through a representation theorem, which allows us to prove that this completion is a completely distributive algebraic lattice. We show several properties about this completion, and we present a connection with the free distributive lattice extension of a distributive nearlattice. Finally, we consider how can be extended n-ary operations on distributive nearlattices, and we study the basic properties of these extensions.
We prove global well-posedness and scattering for the defocusing, cubic NLS on $\mathbb{R}^3$ with initial data in $H^s(\mathbb{R}^3)$ for $s>49/74$. The proof combines the ideas of resonance decomposition in \cite{CKSTT4} and linear-nonlinear decomposition in \cite{ben1}\cite{roy} together with the idea of large time iteration.