论文标题
无单位的Frobenius量子
Unitless Frobenius quantales
论文作者
论文摘要
通常说Frobenius量子一定是Unitital。通过将否定为原始操作，我们可以定义可能没有单位的Frobenius量化。我们发展了这些结构的基本理论，并特别显示了如何定义其商为Frobenius量化的核。通过相位量化，这会产生相位语义和表示定理。这些结构的重要示例来自Raney的紧密galois连接概念：完整晶格的紧密内粒总是形成Girard量子，并且仅当晶格完全分布时，这是Unital的。我们给出了钻石晶格MN的紧密内粒的表征和枚举，并在这些地图上说明了Frobenius结构。通过相位语义，我们展示了从无限尺寸希尔伯特空间上的微量类操作员构建的类似示例。最后，我们认为无法正确添加单位：frobenius量化：Unital Ventalale的所有可能延伸都无法保留否定。
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.
