Atoms, Primes and Implicative Lattices

Abstract

Abstract Let L be an a -implicative semilattice. We obtain a characterization of those elements which cover a . This gives a characterization of atoms in pseudocomplemented semilattices, and leads to various results on primes and irreducibles in semilattices. As an application, we prove that in a complete, atomistic lattice L , the following are equivalent (i) L is implicative (ii) L is (2, ∞) meet distributive (iii) each element of L is a meet of primes.