  In this paper, we show that marked quantales have a reflection into
quantales. To obtain the reflection we construct free quantales over marked
quantales using appropriate lower sets.
  A marked quantale is a posemigroup in which certain admissible subsets are
required to have joins, and multiplication distributes over these. Sometimes
are the admissible subsets in question specified by means of a so-called
selection function. A distinguishing feature of the study of marked quantales
is that a small collection of axioms of an elementary nature allows one to do
much that is traditional at the level of quantales. The axioms are sufficiently
general to include as examples of marked quantales the classes of posemigroups,
$\sigma$-quantales, prequantales and quantales. Furthermore, we discuss another
reflection to quantales obtained by the injective hull of a posemigroup.
