Partial orders, Hasse diagrams, and lattices (intro)

Not every poset is a lattice. A lattice is a poset with additional algebraic structure: the guarantee that every pair of elements has both a least upper bound (join) and a greatest lower bound (meet). The canonical example is the power set lattice (𝒫(S), ⊆) where the join is set union and the meet is set intersection. Another fundamental example is the divisibility lattice on positive integers, where the join of a and b is lcm(a, b) and the meet is gcd(a, b). The diagram below illustrates the power set lattice on a three-element set.

Contrast this with a poset that is not a lattice. Consider the set {a, b, c, d} with a ≤ c, a ≤ d, b ≤ c, b ≤ d, and no other non-trivial relations. The elements c and d are both upper bounds of {a, b}, but neither c ≤ d nor d ≤ c, so there is no least upper bound. The join a ∨ b does not exist, and the structure fails to be a lattice. Recognizing when a poset does or does not satisfy the lattice property is a fundamental skill in discrete mathematics and theoretical computer science.

Tip: A quick test: if you can find any pair of elements whose set of upper bounds has no unique minimum, the poset is not a lattice. Conversely, a finite poset is a lattice if and only if it has a unique maximum, a unique minimum, and every pair has both a join and a meet. In practice, examining the Hasse diagram for a "diamond" or "pentagon" sub-configuration often reveals non-lattice structure.