论文标题
晶格，光谱空间和闭合操作
Lattices, Spectral Spaces, and Closure Operations on Idempotent Semirings
论文作者
论文摘要
由Hochster引入的光谱空间是拓扑空间的同构，是通勤环的主要光谱。在本文中，我们研究了光谱空间的透视范围，因为它是代数结构，因为它在热带几何形状上的多种应用，因此受到了很多关注。我们首先证明一个空间是光谱，并且仅当它是diadempotent semiring的\ emph {prime $ k $ -spectrum}时。实际上，我们通过构建对光谱空间类别的质量的半掌握半模式的子类别来丰富Hochster的定理。我们进一步提供了由半段的一致关系引起的光谱空间的示例。特别是，我们证明了\ emph {估值空间}和\ emph {prime Alloce of Prime Alloce}在一个基本的半静态中是光谱，并且两者之间有自然的集合；这显示了戒指和势力半肌之间的明显差异。然后，我们开发出换向代数的几个方面。我们主要关注\ emph {闭合操作}的概念，并提供了几个示例。特别是，我们引入了一个\ emph {Integral闭合操作}和iDempotent semirings的\ emph {frobenius闭合操作}。
Spectral spaces, introduced by Hochster, are topological spaces homeomorphic to the prime spectra of commutative rings. In this paper we study spectral spaces in perspective of idempotent semirings which are algebraic structures receiving a lot of attention due to its several applications to tropical geometry. We first prove that a space is spectral if and only if it is the \emph{prime $k$-spectrum} of an idempotent semiring. In fact, we enrich Hochster's theorem by constructing a subcategory of idempotent semirings which is antiequivalent to the category of spectral spaces. We further provide examples of spectral spaces arising from sets of congruence relations of semirings. In particular, we prove that the \emph{space of valuations} and the \emph{space of prime congruences} on an idempotent semiring are spectral, and there is a natural bijection of sets between the two; this shows a stark difference between rings and idempotent semirings. We then develop several aspects of commutative algebra of semirings. We mainly focus on the notion of \emph{closure operations} for semirings, and provide several examples. In particular, we introduce an \emph{integral closure operation} and a \emph{Frobenius closure operation} for idempotent semirings.
