190 research outputs found
Studying the inertias of LCM matrices and revisiting the Bourque-Ligh conjecture
Let be a finite set of distinct positive integers.
Throughout this article we assume that the set is GCD closed. The LCM
matrix of the set is defined to be the matrix with
as its element. The famous Bourque-Ligh conjecture
used to state that the LCM matrix of a GCD closed set is always invertible,
but currently it is a well-known fact that any nontrivial LCM matrix is
indefinite and under the right circumstances it can be even singular (even if
the set is assumed to be GCD closed). However, not much more is known about
the inertia of LCM matrices in general. The ultimate goal of this article is to
improve this situation. Assuming that is a meet closed set we define an
entirely new lattice-theoretic concept by saying that an element
generates a double-chain set in if the set
can be expressed as a union of
two disjoint chains (here the set consists of all the elements of
the set that are covered by and is the
smallest meet closed subset of that contains the set ). We then
proceed by studying the values of the M\"obius function on sets in which every
element generates a double-chain set and use the properties of the M\"obius
function to explain why the Bourque-Ligh conjecture holds in so many cases and
fails in certain very specific instances. After that we turn our attention to
the inertia and see that in some cases it is possible to determine the inertia
of an LCM matrix simply by looking at the lattice-theoretic structure of
alone. Finally, we are going to show how to construct LCM matrices in
which the majority of the eigenvalues is either negative or positive
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