= Lattice theory
<Lattice theory> studies partially ordered sets in which every pair has a <greatest lower bound> and a <least upper bound>, called its meet $x\wedge y$ and join $x\vee y$. These operations encode the order: $x\leq y$ exactly when $x\wedge y=x$.
The <power set> of a set, ordered by inclusion, forms a lattice with intersection as meet and union as join. Positive integers ordered by divisibility form another lattice: meet is the <greatest common divisor> and join is the <least common multiple>. Thus the join of two and three is six, while neither number precedes the other in this order. Lattice theory compares such structures through their algebraic identities and order properties.
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