Order scattered distributive continuous lattices are topologically scattered. (English) Zbl 0595.06011

A partially ordered set is said to be order scattered if it contains no order-dense chains; a topological space is said to be topologically scattered if each subspace has at least one isolated point in the relative topology. An order-dense chain in a compact partially ordered space contains a compact perfect subspace (namely the set of limit points of the chain - not the set the author suggests in Lemma 1.1); hence any compact partially ordered space which is topologically scattered must be order scattered. The author’s main result is that the converse holds if the compact ordered space is a distributive continuous lattice equipped with the \(\lambda\)-topology (the unique topology making it a compact ordered space and a topological semilattice). The work was inspired by a question posed by Maurice Pouzet at the 1981 NATO Symposium on Ordered Sets; he had shown the equivalence for the lattice of lower sets of an ordered set and asked if it held for more general classes of lattices.
The proofs are mainly self-contained and rely on building an interesting dichotomy between distributive lattices of locally finite breadth and those that admit a complete semilattice embedding of \(2^ N\). In the last section it is shown that for a distributive continuous lattice of locally finite breadth to be order scattered (equivalently topologically scattered) it is necessary and sufficient that the spectrum of prime elements be order scattered (or topologically scattered in the relative topology).
Reviewer: J.D.Lawson


06B30 Topological lattices
06D05 Structure and representation theory of distributive lattices
54F05 Linearly ordered topological spaces, generalized ordered spaces, and partially ordered spaces
06A12 Semilattices