AI 中文总结
研究满足特定性质的有限生成群，证明其满足短字典序同行旅行者证伪性质及相关语言是正则语言，同时仿照他人例子证明陈述逆命题不成立。
AI 中文摘要
我们证明，满足同行旅行者证伪性质的有限生成群\(G = \langle\Sigma\rangle\)也满足短字典序同行旅行者证伪性质。这意味着存在有限多个短字典序锥类型，因此，相应的短字典序代表语言\(\text{SL}_G\subset (\Sigma\cup\Sigma^{-1})^*\)是正则语言。仿照M. Elder的例子，我们证明上述陈述的逆命题不成立。
英文摘要
We show that a finitely generated group $G=\langleΣ\rangle$ satisfying the falsification by fellow traveller property also satisfies the shortlex falsification by fellow traveller property. This implies that there are finitely many shortlex cone types and, therefore, that the corresponding language of shortlex representatives $\text{SL}_G\subset (Σ\cupΣ^{-1})^*$ is a regular language. Following the example of M. Elder, we prove that the converse of the previous statements do not hold.
