A variety of lattices admits a meet dense regular completion if every lattice in the variety can be embedded into a complete lattice in the variety by an embedding that is meet dense and regular (preserves existing joins and meets). We show that exactly two varieties of lattices admit a meet dense regular completion, the variety of one-element lattices and the variety of all lattices. This extends an earlier result of Harding[8] showing these are the only two varieties of lattices closed under MacNeille completions.
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