Atoms, Primes and Implicative Lattices

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.

Quasi-Implicative Lattices and the Logic of Quantum Mechanics

Mittelstaedt has defined the class of quasi-implicative lattices and shown that an ortholattice (orthocomplemented lattice) is quasi-implicative exactly if it is orthomodular (quasi-modular). He has also shown that the quasi-implication operation is uniquely determined by the quasi-implicative conditions. One of Mittelstaedt's conditions, however, seems to lack immediate intuitive motivation. Consequently, this paper seeks to provide a number of reformulations of the quasi-implicative conditions which are more intuitively plausible. Three sets of conditions are examined, and it is shown that each set of conditions is both necessary and sufficient to ensure that an ortholattice is orthomodular, and each set of conditions uniquely specifies the implication operation to be Mittelstaedt's quasi-implication. Various properties of the quasi-implication are then investigated. In particular, it is shown that the quasi-implication fails to satisfy a number of laws associated with the classical material conditional. Various weakenings of these laws, satisfied by the quasi-implication, are also discussed