Topologies on lattice ordered groups, separation from closed downward sets and conjugations of type Lau

We extend the results of Rubinov and Glover on the separation of normal subsets of (0, + ∞)ⁿ, and those of Martínez-Legaz, Rubinov and Singer on the separation of downward subsets of ℝⁿ, to the unifying framework of subsets of Aⁿ, where A is a continuous conditionally complete lattice ordered group. We also extend earlier results of the second author on the characterizations of conjugations of type Lau of functions f : ℝⁿ → ℝⁿ, to the framework of functions f : Aⁿ → Ā, where Ā is the canonical enlargement of A. These extensions are based on the notions and results of the theory of continuous lattices, generalized in an earlier work of the first author and here, to the conditionally complete case, in particular the way below order relation ≪, and the Scott and Lawson topologies. We also introduce the "way above order relation ≫", and consider the "bi-Scott topology" on A, introduced previously by the first author, which under our continuity assumption on A, turns out to be equal to the order topology; moreover, we show that for this topology, (A, ⊗) is a topological group and (A, ≤) is a topological lattice.