2型模糊真假值代数的Topos观点
A topos view of the type-2 fuzzy truth value algebra
摘要：
众所周知，模糊集理论可以被看作是发生在一个topos。构造这种拓扑有几种等价的方法，一种是作为拓扑空间$Y=[0,1)$上的étalé空间的拓扑，它具有较低的拓扑.在这个拓扑中，集合$X$的模糊子集是常拓扑$X\times Y$的子拓扑，其中$X$具有离散拓扑.本文证明了2型模糊真假值代数与1型模糊真假值代数$\mathfrak{I}=([0,1],\wedge,\vee,\neg,0,1)$的常关系元的子代数所构成的复代数同构.更一般地，我们证明了如果$L$是拓扑空间$Y$的开集格，$\mathfrak{X}$是关系结构，则卷积代数$L^\mathfrak{X}$同构于由$Y$上的关系空间拓扑中的$\mathfrak{X}$给出的常关系空间的子代数构成的复代数.
英文摘要：
It is known that fuzzy set theory can be viewed as taking place within a topos. There are several equivalent ways to construct this topos, one is as the topos of étalé spaces over the topological space $Y=[0,1)$ with lower topology. In this topos, the fuzzy subsets of a set $X$ are the subobjects of the constant étalé $X\times Y$ where $X$ has the discrete topology. Here we show that the type-2 fuzzy truth value algebra is isomorphic to the complex algebra formed from the subobjects of the constant relational étalé given by the type-1 fuzzy truth value algebra $\mathfrak{I}=([0,1],\wedge,\vee,\neg,0,1)$. More generally, we show that if $L$ is the lattice of open sets of a topological space $Y$ and $\mathfrak{X}$ is a relational structure, then the convolution algebra $L^\mathfrak{X}$ is isomorphic to the complex algebra formed from the subobjects of the constant relational étalé given by $\mathfrak{X}$ in the topos of étalé spaces over $Y$.
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