Join and meet

In a partially ordered set, the join of elements is their least upper bound, and their meet is their greatest lower bound, when these bounds exist.

Connect

Partially ordered set: A set equipped with a reflexive, antisymmetric, transitive relation. Joins and meets are defined relative to this order relation.
Divisibility: The relation in which one integer divides another without remainder. For positive integers ordered by divisibility, join and meet are least common multiple and greatest common divisor.
Lattice (order theory): A partially ordered set in which every pair of elements has a join and a meet. A lattice is precisely an order where pairwise joins and meets always exist.
Maximal element: An element with no strictly greater element in a partially ordered set. A maximal upper bound need not be least, so it need not be a join.
Upper bound: An element greater than or equal to every element in a specified subset of a partially ordered set. A join is the least element among the upper bounds.
Least common multiple: The smallest positive integer divisible by each of two or more given integers. Under divisibility, it is the join of the given integers.
Boolean algebra: An algebraic structure with meet, join, and complement operations satisfying the Boolean laws. Logical conjunction and disjunction behave as meet and join in its order.
Minimal element: An element with no strictly smaller element in a partially ordered set. A minimal lower bound need not be greatest, so it need not be a meet.
Lower bound: An element less than or equal to every element in a specified subset of a partially ordered set. A meet is the greatest element among the lower bounds.
Greatest common divisor: The greatest positive integer that divides each of two or more given integers. Under divisibility, it is the meet of the given integers.