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.

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.