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.

翻译：经常有人说,Frobenius 方圆形必须是单向的。 通过将否定作为原始操作, 我们可以定义可能没有单位的方圆形。 我们开发了这些结构的基本理论, 并特别展示了如何定义Frobenius 方圆形的核。 这产生了一个阶段的语义和通过阶段四方圆形的表示论。 这些结构的重要例子来自Raney的紧紧加加罗洛丝连接的概念: 完整的拉蒂斯的紧紧内衣总是形成一个Girard 方圆形, 只有拉蒂斯是完全分散的, 才会形成一个单一的。 我们给出了这些图中方圆形方圆形的严格内衣的特征和插图, 并展示了这些图中的Frobenius结构。 通过阶段的语义学, 我们展示了类似的例子, 是在无限的维度的Hilbert空间上从痕量级级操作者所建立起来的类似的例子。 最后, 我们论证说, 单位不能被恰当地添加到Frobenius 方圆形方圆形的方圆形: 方圆形的每一种都无法保存。