Join-irreducible element

Summary

An element of a lattice is join-irreducible if it is not the bottom element and not the join of two smaller elements. In a finite lattice, every element is a join of join-irreducible elements, so these elements generate the lattice by joins. For finite distributive lattices, the poset of join-irreducible elements determines the whole lattice, which is the content of Birkhoff's representation theorem. Meet-irreducible elements are defined dually.