Partial orders, Hasse diagrams, and lattices (intro)

A partial order is a binary relation on a set that is reflexive, antisymmetric, and transitive. The pair (S, ≤) is called a poset. Unlike a total order, a poset permits incomparable elements—pairs where neither a ≤ b nor b ≤ a. A Hasse diagram visualizes a finite poset by plotting elements at vertical positions reflecting the order, drawing edges only for covering relations (a ≺ b: a < b with nothing strictly between), and omitting transitive and reflexive edges.

A lattice is a poset in which every pair of elements possesses both a join (least upper bound, ∨) and a meet (greatest lower bound, ∧). Key examples include the power set lattice (𝒫(S), ⊆) with join = union and meet = intersection, and the divisibility lattice with join = lcm and meet = gcd. Every finite lattice has a unique top element ⊤ and bottom element ⊥. These structures are foundational in algebra, logic, combinatorics, and computer science—from circuit design to compiler optimization to formal verification.