Unitless Frobenius Quantales

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