A topos view of the type-2 fuzzy truth value algebra

John Harding; Carol Walker

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arXiv · 1810.07565

A topos view of the type-2 fuzzy truth value algebra

John Harding·Carol Walker

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Abstract

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 \'{e}tal\'{e} 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 \'{e}tal\'{e} $X\times Y$ 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 \'{e}tal\'{e} 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 \'{e}tal\'{e} given by $\mathfrak{X}$ in the topos of \'{e}tal\'{e} spaces over $Y$.

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BibTeX RIS

John Harding, Carol Walker. 2018-10-17. A topos view of the type-2 fuzzy truth value algebra. https://arxiv.org/abs/1810.07565

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