One of the fundamental notions of linear algebra is the concept of a basis for a vector space. In the category theoretic formulation of universal algebra, vector spaces are the Eilenberg-Moore algebras over the free vector space monad on the category of sets. In this paper we show that the notion of a basis can be extended to algebras of arbitrary monads on arbitrary categories. On the one hand, we establish purely algebraic results, for instance about the existence and uniqueness of bases and the representation of algebra morphisms. On the other hand, we use the general notion of a basis in the context of coalgebraic systems and show that a basis for the underlying algebra of a bialgebra gives rise to an equivalent free bialgebra. As a result, we are, for instance, able to recover known constructions from automata theory, namely the so-called canonical residual finite state automaton. Finally, we instantiate the framework to a variety of example monads, including the powerset, downset, distribution, and neighbourhood monad.

中文翻译：

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monad 上的代数基础

线性代数的基本概念之一是向量空间的基概念。在泛代数的范畴论表述中，向量空间是在集合范畴上的自由向量空间单子上的 Eilenberg-Moore 代数。在本文中，我们表明基的概念可以扩展到任意类别上的任意 monad 的代数。一方面，我们建立纯代数结果，例如关于基的存在性和唯一性以及代数态射的表示。另一方面，我们在余代数系统的上下文中使用基的一般概念，并表明双代数的基础代数的基产生等效的自由双代数。结果，例如，我们能够从自动机理论中恢复已知的结构，即所谓的规范残差有限状态自动机。最后，我们将框架实例化为各种示例 monad，包括 powerset、downset、分布和邻域 monad。