  This paper studies many-to-one assignment markets, or matching markets with
wages. Although it is well-known that the core of this model is non-empty, the
structure of the core has not been fully investigated. To the known
dissimilarities with the one-to-one assignment game, we add that the bargaining
set does not coincide with the core and the kernel may not be included in the
core. Besides, not all extreme core allocations can be obtained by means of a
lexicographic maximization or a lexicographic minimization procedure, as it is
the case in the one-to-one assignment game.
  The maximum and minimum competitive salaries are characterized in two ways:
axiomatically and by means of easily verifiable properties of an associated
directed graph. Regarding the remaining extreme core allocations of the
many-to-one assignment game, we propose a lexicographic procedure that, for
each order on the set of workers, sequentially maximizes or minimizes each
worker's competitive salary. This procedure provides all extreme vectors of
competitive salaries, that is all extreme core allocations.
