  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.
