What structure can be placed on the burgeoning field of fractional calculus with assorted kernel functions? This question has been addressed by the introduction of various general kernels, none of which has both a fractional order parameter and a clear inversion relation. Here, we use ideas from abstract algebra to construct families of fractional integral and derivative operators, parametrised by a real or complex variable playing the role of the order. These have the typical behaviour expected of fractional calculus operators, such as semigroup and inversion relations, which allow fractional differential equations to be solved using operational calculus in this general setting, including all types of fractional calculus with semigroup properties as special cases.

具有各种核函数的分数阶微积分新兴领域可以放置什么结构？这个问题已经通过引入各种通用核来解决，但没有一个核同时具有分数阶参数和明确的反演关系。在这里，我们使用抽象代数的思想来构建分数积分和导数算子族，并通过扮演阶数角色的实数或复数变量进行参数化。它们具有分数阶微积分算子的典型行为，例如半群和反演关系，允许在这种一般设置中使用运算微分方程求解分数阶微分方程，包括作为特殊情况的具有半群属性的所有类型的分数阶微积分。