论文标题
半频谱
Quantalic spectra of semirings
论文作者
论文摘要
频谱构造出现在整个数学过程中，是从代数数据构建拓扑空间的一种方式。考虑到通勤性的局部半度r（拓扑半程的无点类似物），我们定义了R频谱，该频谱概括了分布晶格的石材光谱，通勤环的Zariski频谱，gelfantative underative c*-allital c*-Algebra and Hofmann-law septrumss spectrums septrum septrum然后，我们在R上的条件下为该频谱提供了明确的结构，这是我们的主要示例所满足的。 我们的结果是建设性的，因此在任何具有自然数对象的基本拓扑基础上接受解释。因此，我们构建的频谱实际上应该是一个位置，而不是拓扑空间。 对我们的构造的简单修改会以换向数量的形式产生怪异的频谱。除本地频谱的纯粹拓扑信息外，这种量化还包含“差异”信息。在离散环的情况下，我们的构造产生了理想的量化。 这促使我们更详细地研究了理想的量化。我们讨论了量化量的抽象理想理论的一些结果，并为量化以与交换环理论类似的方式提供了量子的含义提供了初步定义。
Spectrum constructions appear throughout mathematics as a way of constructing topological spaces from algebraic data. Given a commutative localic semiring R (the pointfree analogue of a topological semiring), we define a spectrum of R which generalises the Stone spectrum of a distributive lattice, the Zariski spectrum of a commutative ring, the Gelfand spectrum of a commutative unital C*-algebra and the Hofmann-Lawson spectrum of a continuous frame. We then provide an explicit construction of this spectrum under conditions on R which are satisfied by our main examples. Our results are constructively valid and hence admit interpretation in any elementary topos with natural number object. For this reason the spectrum we construct should actually be a locale instead of a topological space. A simple modification to our construction gives rise to a quantic spectrum in the form of a commutative quantale. Such a quantale contains `differential' information in addition to the purely topological information of the localic spectrum. In the case of a discrete ring, our construction produces the quantale of ideals. This prompts us to study the quantale of ideals in more detail. We discuss some results from abstract ideal theory in the setting of quantales and provide a tentative definition for what it might mean for a quantale to be nonsingular by analogy to commutative ring theory.
