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 \( d = \prod\nolimits_{k = 1}^r {p_k^{d^{(k)} } } \) is said to be an exponential divisor or an e-divisor of \( n = \prod\nolimits_{k = 1}^r {p_k^{n^{(k)} } } \) (n > 1), written as d | e n, if d (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.
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Dedicated to the memory of Professor M V Subbarao
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Korkee, I., Haukkanen, P. Meet and join matrices in the poset of exponential divisors. Proc Math Sci 119, 319–332 (2009). https://doi.org/10.1007/s12044-009-0031-2
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DOI: https://doi.org/10.1007/s12044-009-0031-2