标题： 存在与回归：重新审视连乘在梅塔-列弗勒函数中的失败 显示英文标题

标题： There and Back Again: Revisiting the Failure of Concatenation in Mittag-Leffler Functions

摘要： 函数$t \mapsto E_{\alpha}(\lambda t^\alpha)$被广泛认为是指数函数的分数阶类似物，但其代数性质仍然不为人所理解。 特别是，标准参考文献中缺乏对半群性质失败的严格证明。 在本工作中，我们通过提供一个分析证明来填补这一空白，证明半群性质成立当且仅当$\alpha = 1$或$\lambda = 0$。 我们的结果确立了 Mittag-Leffler 动力学与半群演化相容的精确条件，从而强调了分数阶微分方程的内在非局部性。 显示英文摘要

摘要： The function $t \mapsto E_{\alpha}(\lambda t^\alpha)$ is widely regarded as the fractional analogue of the exponential function, yet its algebraic properties remain poorly understood. In particular, standard references lack a rigorous proof of the failure of the semigroup property. In this work, we fill this gap by providing an analytical proof that the semigroup property holds if and only if either $\alpha = 1$ or $\lambda = 0$. Our result establishes a precise criterion for when Mittag-Leffler dynamics are compatible with semigroup evolution, thereby emphasizing the intrinsic nonlocality of fractional-order differential equations.