  It is often stated that Frobenius quantales are necessarily unital. By taking
negation as a primitive operation, we can define Frobenius quantales that may
not have a unit. We develop the elementary theory of these structures and show,
in particular, how to define nuclei whose quotients are Frobenius quantales.
This yields a phase semantics and a representation theorem via phase quantales.
Important examples of these structures arise from Raney's notion of tight
Galois connection: tight endomaps of a complete lattice always form a Girard
quantale which is unital if and only if the lattice is completely distributive.
We give a characterisation and an enumeration of tight endomaps of the diamond
lattices Mn and exemplify the Frobenius structure on these maps. By means of
phase semantics, we exhibit analogous examples built up from trace class
operators on an infinite dimensional Hilbert space. Finally, we argue that
units cannot be properly added to Frobenius quantales: every possible extention
to a unital quantale fails to preserve negations.
