DISTRIBUTIVE PAIRS IN BIATOMIC LATTICES
B. N. Waphare, Vinayak V. Joshi · 2004
We prove that a biatomic lattice L is distributive if and only if every pair of atoms of L is distributive. This result has been used to obtain characterizations of distributive pairs in terms of semi-distributive pairs, del-relation and perspectivity. In an atomistic lattice (every non-zero element is the join of atoms contained in it) L, for a pair of non-zero elements a,b 2 L we write (a,b)P, if for every atom p a _ b there exist atoms q, r such that p q _ r, q a and r b. L is called biatomic if (a,b)P holds for all non-zero elements a,b 2 L. In (2), Bennett studied the class of biatomic and provided many impor- tant examples. In fact, the same class with the nomenclature additive lattices is also studied by Bennett (1). Biatomic are also defined in terms of P- relation.