Characterization of lattices by unique quasi-suprema and quasi-infima

Let a partially ordered family be a family equipped with a partial order. For two elements, a quasi-supremum is a minimal upper bound and a quasi-minimum is a maximal lower bound. Lattice characterization conjecture. A partially ordered family is a lattice if and only if each pair of its elements has a unique quasi-supremum and a unique quasi-minimum.

This asks whether the existence and uniqueness of quasi-suprema and quasi-infima for every pair characterizes lattices. The forward implication follows from the definitions of supremum and infimum; the converse is presented as an open question in the source.