论文标题
分布晶格，关联几何形状：算术案例
Distributive lattices, associative geometries: the arithmetic case
论文作者
论文摘要
我们证明了五个论点的身份，在与GCD和LCM作为晶格操作的自然数字中有效。更普遍地，这种身份是任意分布晶格的特征。通过修复五个论点中的三个，我们始终获得关联产品，因此每个分布晶格都带有许多半群结构。在算术情况下，我们明确计算了此类半群的乘法表，并描述了它们的某些特性。它们中的许多是周期性的，可以看作是环z/nz的“非共同类似物”。
We prove an identity for five arguments, valid in the lattice of natural numbers with gcd and lcm as lattice operations. More generally, this identity characterizes arbitrary distributive lattices. Fixing three of the five arguments, we always get associative products, and thus every distributive lattice carries many semigroup structures. In the arithmetic case, we explicitly compute multiplication tables of such semigroups and describe some of their properties. Many of them are periodic, and can be seen as "non-commutative analogs" of the rings Z/nZ.
