Weihrauch度的代数结构
On the algebraic structure of Weihrauch degrees
AI总结：
本文在Weihrauch度上引入复合积与蕴含两种新运算，研究其代数结构、分配律、理想与商，并借助多值连续函数空间进行刻画。
摘要：
我们在Weihrauch度上引入了两种新的运算（复合积与蕴含），并研究了其整体代数结构。我们研究了各种分配律的有效性，这构成了与类似结构（如剩余格和并发Kleene代数）进行比较的基础。通过引入关于复合积的理想的概念，我们可以考虑Weihrauch度的适当商。我们还利用蕴含证明了一些特定的刻画。为了引入和研究复合积与蕴含，我们引入并研究了一个多值连续函数的函数空间。这个空间对于有效可追踪空间（与可容许表示空间密切相关）表现得特别好。
英文摘要：
We introduce two new operations (compositional products and implication) on Weihrauch degrees, and investigate the overall algebraic structure. The validity of the various distributivity laws is studied and forms the basis for a comparison with similar structures such as residuated lattices and concurrent Kleene algebras. Introducing the notion of an ideal with respect to the compositional product, we can consider suitable quotients of the Weihrauch degrees. We also prove some specific characterizations using the implication. In order to introduce and study compositional products and implications, we introduce and study a function space of multi-valued continuous functions. This space turns out to be particularly well-behaved for effectively traceable spaces that are closely related to admissibly represented spaces.
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