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.
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