  The present paper is concerned with the various algebraic structures
supported by the set of Tur\'an densities.
  We prove that the set of Tur\'an densities of finite families of r-graphs is
a non-trivial commutative semigroup, and as a consequence we construct explicit
irrational densities for any r >= 3. The proof relies on a technique recently
developed by Pikhurko.
  We also show that the set of all Tur\'an densities forms a graded ring, and
from this we obtain a short proof of a theorem of Peng on jumps of hypergraphs.
  Finally, we prove that the set of Tur\'an densities of families of r-graphs
has positive Lebesgue measure if and only if it contains an open interval. This
is a simple consequence of Steinhaus's theorem.
