It seems like the existence of a Galois connection between \$$(A,\leq_A) \$$ and \$$(B, \leq_B ) \$$ requires least-upper bounds and greatest-lower bounds to be defined. Does a Galois connection between \$$(A,\leq_A) \$$ and \$$(B, \leq_B ) \$$ imply that \$$(A,\leq_A) \$$ and \$$(B, \leq_B ) \$$ are complete lattices?

For complete lattices, LUB and GLB need to exist for arbitrary subsets, but for a Galois connection, the subsets are not arbitrary. Yet, I haven't been able to formulate an example of a Galois connection between posets that are not complete lattices.