Topology of closure systems in algebraic lattices
Niels Schwartz · Algebra Universalis · 2023
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 $$\langle X \rangle _P$$ ⟨ X ⟩ P . The closure system $$\langle Y \rangle _P$$ ⟨ Y ⟩ P generated by the patch closure Y of X is the patch closure of $$\langle X \rangle _P$$ ⟨ X ⟩ P . If X is contained in the set of nontrivial prime elements of P then $$\langle X \rangle _P$$ ⟨ X ⟩ 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.