Topology of closure systems in algebraic lattices

Abstract

Algebraic lattices are spectral spaces for the coarse lower topology. Closure systems in algebraic lattices are studied as subspaces. Connections between order theoretic properties of a closure system and topological properties of the subspace are explored. A closure system is algebraic if and only if it is a patch closed subset of the ambient algebraic lattice. Every subset X in an algebraic lattice P generates a closure system \(\langle X \rangle _P\). The closure system \(\langle Y \rangle _P\) generated by the patch closure Y of X is the patch closure of \(\langle X \rangle _P\). If X is contained in the set of nontrivial prime elements of P then \(\langle X \rangle _P\) is a frame and is a coherent algebraic frame if X is patch closed in P. Conversely, if the algebraic lattice P is coherent then its set of nontrivial prime elements is patch closed.