It is well known that the meet-closed subsets of a partially ordered set P, ordered by set inclusion, form a compactly generated, dually compactly generated, distributive lattice, which we denote by LM(P). Moreover, LM(P ) is atomic if and only if P satisfies the Descending Chain Condition. In this paper we define a closure relation D--, D on LM(P ) in terms of an equivalence relation 0 on P satisfying specified conditions whereby the bar-closed sets of L~(P) form an upper locally distributive lattice LMo(P ). We thereby obtain an "upper distributive" theory that corresponds to the theory for LM(P ).