A dcpo $P$ is continuous if and only if the lattice $C\left(P\right)$ of all Scott-closed subsets of $P$ is completely distributive. However, in the case where $P$ is a non-continuous dcpo, little is known about the order structure of $C\left(P\right)$. In this paper, we study the order-theoretic properties of $C\left(P\right)$ for general dcpo’s $P$. The main results are: (i) every $C\left(P\right)$ is C-continuous; (ii) a complete lattice $L$ is isomorphic to $C\left(P\right)$ for a complete semilattice $P$ if and only if $L$ is weak-stably C-algebraic; (iii) for any two complete semilattices...

Locally solid Riesz spaces have been widely investigated in the past several decades; but locally solid topological lattice-ordered groups seem to be largely unexplored. The paper is an attempt to initiate a relatively systematic study of locally solid topological lattice-ordered groups. We give both Roberts-Namioka-type characterization and Fremlin-type characterization of locally solid topological lattice-ordered groups. In particular, we show that a group topology on a lattice-ordered group is...

It is proved that for every continuous lattice there is a unique semiuniform structure generating both the order and the Lawson topology. The way below relation can be characterized with this uniform structure. These results are used to extend many of the analytical properties of real-valued l.s.cḟunctions to l.s.cḟunctions with values in a continuous lattice. The results of this paper have some applications in potential theory.