回转群上的Fuzzy回转范数
Fuzzy gyronorms on gyrogroups
摘要：
回转群的概念是没有明确结合性的群的推广。本文引入了回转群上模糊回转范数的概念。研究了模糊度量（George和Veeramani意义下）、模糊回转范和回转群上的回转范之间的关系。并讨论了模糊赋范回转群上的模糊度量结构。最后，研究了具有不变度量的回转群的模糊度量完备化。本文主要证明了：若$d$是回转群$G$上的不变度量，$(\widehat{G},\widehat{d})$是度量空间$(G,d)$的度量完备，则对任意连续的$t$ -范数$\ast$，$(\widehat{G},\widehat{d})$的标准模糊度量空间$(\widehat{G},M_{\widehat{d}},\ast)$是度量空间$(G,d)$的度量完备，则$(\widehat{G},\widehat{d})$是度量空间$(G,d)$的度量完备。$(G,d)$的标准模糊度量空间$(G,M_d,\ast)$的唯一模糊度量完备化;此外，$(\widehat{G},M_{\widehat{d}},\ast)$是包含$(G,M_d,\ast)$作为稠密模糊度量子回转群的模糊度量回转群，并且$M_{\widehat{d}}$在$\widehat{G}$上是不变的。应用这个结果，我们得到了每个具有不变度量$d$的回转群$G$都有一个$(G,d)$的完备度量空间$(\widehat{G},\widehat{d})$，使得由$\widehat{d}$引入拓扑的$\widehat{G}$是一个拓扑回转群，其中包含$G$作为稠密子回转群，并且$\widehat{d}$在$\widehat{G}$上不变.
英文摘要：
The concept of gyrogroups is a generalization of groups which do not explicitly have associativity. In this paper, the notion of fuzzy gyronorms on gyrogroups is introduced. The relations of fuzzy metrics (in the sense of George and Veeramani), fuzzy gyronorms and gyronorms on gyrogroups are studied. Also, the fuzzy metric structures on fuzzy normed gyrogroups are discussed. In the last, the fuzzy metric completion of a gyrogroup with an invariant metric are studied. We mainly show that let $d$ be an invariant metric on a gyrogroup $G$ and $(\widehat{G},\widehat{d})$ is the metric completion of the metric space $(G,d)$; then for any continuous $t$-norm $\ast$, the standard fuzzy metric space $(\widehat{G},M_{\widehat{d}},\ast)$ of $(\widehat{G},\widehat{d})$ is the (up to isometry) unique fuzzy metric completion of the standard fuzzy metric space $(G,M_d,\ast)$ of $(G,d)$; furthermore, $(\widehat{G},M_{\widehat{d}},\ast)$ is a fuzzy metric gyrogroup containing $(G,M_d,\ast)$ as a dense fuzzy metric subgyrogroup and $M_{\widehat{d}}$ is invariant on $\widehat{G}$. Applying this result, we obtain that every gyrogroup $G$ with an invariant metric $d$ admits an (up to isometric) unique complete metric space $(\widehat{G},\widehat{d})$ of $(G,d)$ such that $\widehat{G}$ with the topology introduced by $\widehat{d}$ is a topology gyrogroup containing $G$ as a dense subgyrogroup and $\widehat{d}$ is invariant on $\widehat{G}$.
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