Is the unique lattice on the empty set, also a complete lattice? More precisely, $(\emptyset, \emptyset)$ is the empty ordering, which is also a lattice. Is it also a complete lattice?
2026-04-09 16:33:17.1775752397
Is the empty lattice a complete lattice?
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While the empty lattice is indeed a lattice - vacuously: any two elements have least upper bounds and greatest lower bounds and these behave the way they should - it is not, however, complete: the emptyset has no least upper bound, or greatest lower bound.
Note that complete lattices are bounded: the least upper bound of the emptyset has to be the minimal element, and the greatest lower bound of the emptyset has to be the maximal element.