论文标题
晶格理论方法的一些进一步应用在奇异LCM矩阵的研究中
Some further applications of a lattice theoretic method in the study of singular LCM matrices
论文作者
论文摘要
在1876年，H。J. S. Smith定义了一个LCM矩阵，如下所示：令S = {x_1，x_2，...，...，x_n}是一组正整数。 LCM矩阵[S]是带有LCM（X_I，X_J）的N $ \ times $ n矩阵作为其IJ条目。在过去的30年中，LCM矩阵的奇异性使许多作者感兴趣。在1992年，布尔克（Bourque）和利格（Ligh）最终猜测，如果集合S的GCD闭合度（这意味着GCD（x_i，x_j）$ \ in $ s in $ s in $ s in $ s in $ {$ {1，2，.。。。。。。。。但是，几年后，Haukkanen等人首先证明了这一猜想是错误的。然后是洪。事实证明，猜想仅在GCD封闭套装上，最多有7个元素，但总体上不适合大型集合。但是，给定的反例并未给出太多了解为什么在n = 8的情况下，猜想完全失败。后来在几篇文章中改善了这种情况，其中引入了一种新的晶格理论方法（该方法基于以下事实：由于假定集合s已关闭GCD，因此结构（S，|）实际上形成了遇到半层次的结构）。例如，已经表明，在集合S具有8个元素并且矩阵[s]的情况下，（S，|）的半层次结构（即Cube结构）只有一个选项。 由于在各种文章中对n = 8的病例进行了彻底的研究，因此下一个自然步骤是将方法应用于n = 9的情况。这是由Altinisik和Altintaa考虑的，因为他们考虑了（S，|）的不同晶格结构，其九个元素可以导致奇异的LCM矩阵[S]。但是，他们的调查留下了两个开放的问题，本演讲的主要目的是为他们提供解决方案。我们还将给出一个新的晶格理论证明，以称为太阳的猜想，最初是通过洪理论方法证明的。
In 1876 H. J. S. Smith defined an LCM matrix as follows: let S = {x_1, x_2, ..., x_n} be a set of positive integers. The LCM matrix [S] is the n $\times$ n matrix with lcm(x_i , x_j) as its ij entry. During the last 30 years singularity of LCM matrices has interested many authors. In 1992 Bourque and Ligh ended up conjecturing that if the GCD closedness of the set S (which means that gcd(x_i, x_j) $\in$ S for all i, j $\in$ {1, 2, . . . , n}), suffices to guarantee the invertibility of the matrix [S]. However, a few years later this conjecture was proven false first by Haukkanen et al. and then by Hong. It turned out that the conjecture holds only on GCD closed sets with at most 7 elements but not in general for larger sets. However, the given counterexamples did not give much insight on why does the conjecture fail exactly in the case when n=8. This situation was later improved in a couple of articles, where a new lattice theoretic approach was introduced (the method is based on the fact that because the set S is assumed to be GCD closed, the structure (S, |) actually forms a meet semilattice). For example, it has been shown that in the case when the set S has 8 elements and the matrix [S] is singular, there is only one option for the semilattice structure of (S, |), namely the cube structure. Since the cases up to n=8 have been thoroughly studied in various articles, the next natural step is to apply the methods to the case n=9. This was done by Altinisik and Altintaa as they consider the different lattice structures of (S, |) with nine elements that can result in a singular LCM matrix [S]. However, their investigation leaves two open questions, and the main purpose of this presentation is to provide solutions to them. We shall also give a new lattice theoretic proof for a result referred to as Sun's conjecture, which was originally proven by Hong via number theoretic approach.
