A Characterization for non-DCC Lattices

Join-irreducible elements in a lattice have an important role. They act like blocks of a lattice. In DCC lattices each element of the lattice has a unique ¯nite representation as a join of join-irreducible elements. In this paper, we seek lattices which contains elements that can be represented as an in¯nite supremum of join-irreducible elements. One of these lattices is the lattice of sequences. Finally, we give a new characterization for such lattices.