We provide graded extensions of algebraic theories and Lawvere theories that correspond to graded monads. We prove that graded algebraic theories, graded Lawvere theories, and finitary graded monads are equivalent via equivalence of categories, which extends the equivalence for monads. We also give sums and tensor products of graded algebraic theories to combine computational effects as an example of importing techniques based on algebraic theories to graded monads.

中文翻译：

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分级代数理论

我们提供与分级单子相对应的代数理论和 Lawvere 理论的分级扩展。我们通过范畴的等价证明了分级代数理论、分级 Lawvere 理论和有限分级单子是等价的，这扩展了单子的等价性。我们还给出了分级代数理论的总和和张量积，以结合计算效果作为将基于代数理论的技术导入分级 monad 的示例。