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A 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: 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. [ABSTRACT FROM AUTHOR]

An extension principle for T-spaces is given which includes several well-known compactifications. Moreover, it is shown that the compact T-spaces are exactly the T-retracts of Stone spaces. All these results are derived from some general facts on semirings stated in the first part of the paper. [ABSTRACT FROM AUTHOR]