  Quantum entanglement is known to be monogamous, i.e., it obeys strong
constraints on how the entanglement can be distributed among multipartite
systems. Almost all the entanglement monotones so far are shown to be
monogamous. We explore here a family of entanglement monotones with the reduced
functions are concave but not strictly concave and show that they are not
monogamous. They are defined by four kinds of the ``partial-norm'' of the
reduced state, which we call them \textit{partial-norm of entanglement},
minimal partial-norm of entanglement, reinforced minimal partial-norm of
entanglement, and \textit{partial negativity}, respectively. This indicates
that, the previous axiomatic definition of the entanglement monotone needs
supplemental agreement that the reduced function should be strictly concave
since such a strict concavity can make sure that the corresponding convex-roof
extended entanglement monotone is monogamous. Here, the reduced function of an
entanglement monotone refers to the corresponding function on the reduced state
for the measure on bipartite pure states.
