Catalytic distributive lattices and compact zero-dimensional topological lattices
Let D(0,1) denote the category of bounded distributive lattices and lattice homomorphisms preserving 0 and 1. A lattice L is said to be catalytic if Hom(L,M) is a lattice for all M. It is known from the work of \textit{R. Balbes} [Algebra Univers. 11, 334-340 (1980; Zbl 0457.06008)] that L is catalytic if and only if every element is a finite meet of elements in M(L) (the meet-irreducible elements), M(L) is a join subsemilattice, \(0\in M(L)\), and dual conditions hold for J(L), the join irreducibles. In this paper these conditions are translated to the Priestley dual in the category of compact totally order disconnected spaces. The principal result is that a lattice L is catalytic if and only if its dual is a topological lattice. This result is used to show L is catalytic if and only if it is a retract of a lattice freely generated by an ordered set.
category of bounded distributive lattices
category of compact totally order disconnected spaces
scientific article; zbMATH DE number 3887754
scientific article