Set unions and least common multiples share one hidden mathematical shape
Merging two sets, finding the least common multiple of two numbers and combining two logical conditions look like unrelated chores. Order theorists see the same pattern in each: a lattice, a collection where any two items have a unique smallest thing above both of them and a unique largest thing below both.
A lattice starts from a partial order, a way of saying some elements sit below others without requiring that every pair be comparable. What makes it a lattice is that each pair has a join, or least upper bound, and a meet, or greatest lower bound. For all the subsets of a given set, ordered by inclusion, the join is the union and the meet is the intersection. For the natural numbers ordered by divisibility, the join is the least common multiple and the meet is the greatest common divisor.
The same structure can be defined purely by algebra. Instead of an ordering, you start with two operations that are commutative and associative and obey the absorption laws, the only rules that mention both operations at once. Those laws force the two operations to agree on a single underlying order, and the order can be read back off: one element sits below another when meeting them returns the first. Because the order and algebra versions are equivalent, mathematicians switch between them freely.
Some consequences come almost for free. By induction, every non-empty finite collection in a lattice has a join and a meet. A bounded lattice adds a top element and a bottom element; every non-empty finite lattice is automatically bounded, and any lattice can be made bounded simply by attaching a new top and bottom.
Lattices also sit among more familiar algebraic families. Each of the two operations on its own forms a commutative semigroup, so a lattice is a pair of them sharing one set, glued together by absorption. Notable special kinds include distributive lattices, the Boolean algebras behind classical logic, the Heyting algebras used in intuitionistic reasoning and geometric lattices, also known as matroids.
Source: Lattice (order)
