Standard completions for quasiordered sets

AbstractA standard completion for a quasiordered set Q is a closure system whose point closures are the principal ideals of Q. We characterize the following types of standard completions by means of their closure operators:(i)V-distributive completions,(ii)Completely distributive completions,(iii)A-completions (i.e. standard completions which are completely distributive algebraic lattices),(iv)Boolean completions. Moreover, completely distributive completions are described by certain idempotent relations, and the A-completions are shown to be in one-to-one correspondence with the join-dense subsets of Q. If a pseudocomplemented meet-semilattice Q has a Boolean completion ℭ, then Q must be a Boolean lattice and ℭ its MacNeille completion.