Meet and join matrices in the poset of exponential divisors

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 = Pi(r)(k=1) p(k)(d(k)) is said to be an exponential divisor or an e-divisor of v = Pi(r)(k=1) p(k)(n(k)) (n>1), written as d |(e) n, if d((k)) |n((k)) for all prime divisors pk 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.