Meet and join matrices in the poset of exponential divisors

Abstract

It is well-known that (Z + , |) = (Z + , GCD, LCM) is a lattice, where | is the usual divisibility relation and GCD and LCM stand for the greatest common divisor and the least common multiple of positive integers.The number d = r k=1 p d (k) k is said to be an exponential divisor or an e-divisor of n = r k=1 p n (k) k (n > 1), written as d | e n, if d (k) | n (k) for all prime divisors p k of n. It is easy to see that (Z + \{1}, | e ) is a poset under the exponential divisibility relation but not a lattice, since the greatest common exponential divisor (GCED) and the least common exponential multiple (LCEM) do not always exist.In this paper we embed this poset in a lattice. As an application we study the GCED and LCEM matrices, analogues of GCD and LCM matrices, which are both special cases of meet and join matrices on lattices.