  We introduce (weighted) Segre and Rees products for posets and show that
these constructions preserve the Cohen-Macaulay property over a field $k$ and
homotopically. As an application we show that the weighted Segre product of two
affine semigroup rings that are Koszul is again Koszul. This result generalizes
previous results by Crona on weighted Segre products of polynomial rings.
  We also give a new proof of the fact that the Rees ring of a Koszul affine
semigroup ring is again Koszul.
  The paper ends with a list of some open problems in the area.
