@article{Schwartz2023, author = {Schwartz, Niels}, title = {Topology of closure systems in algebraic lattices}, series = {Algebra universalis}, volume = {84}, journal = {Algebra universalis}, number = {2}, publisher = {Springer Nature}, address = {Berlin}, doi = {10.1007/s00012-023-00815-7}, url = {http://nbn-resolving.de/urn:nbn:de:101:1-2023090808111682406614}, pages = {1 -- 33}, year = {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〈X〉P . The closure system〈Y 〉P generated by the patch closure Y of X is the patch closure of〈X〉P. If X is contained in the set of nontrivial prime elements of P then〈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.}, language = {en} }