Lattice theory studies partially ordered sets in which every pair has a greatest lower bound and a least upper bound, called its meet and join . These operations encode the order: exactly when .
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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Lattice theory is a branch of abstract algebra that studies mathematical structures known as lattices. A lattice is a partially ordered set (poset) in which every two elements have a unique supremum (least upper bound, also known as join) and an infimum (greatest lower bound, also known as meet). ### Key Concepts in Lattice Theory 1.