Motivated by recent work on weak distributive laws and their applications to coalgebraic semantics, we investigate the algebraic nature of semialgebras for a monad. These are algebras for the underlying functor of the monad subject to the associativity axiom alone-the unit axiom from the definition of an Eilenberg-Moore algebras is dropped. We prove that if the underlying category has coproducts, then semialgebras for a monad M are in fact the Eilenberg-Moore algebras for a suitable monad structure on the functor id + M , which we call the semifree monad M^s. We also provide concrete algebraic presentations for semialgebras for the maybe monad, the semigroup monad and the finite distribution monad. A second contribution is characterizing the weak distributive laws of the form M T $\Rightarrow$ T M as strong distributive laws M^s T $\Rightarrow$ T M^s subject to an additional condition.
翻译:受最近关于薄弱分布法及其应用于煤眼语义学的研究的启发,我们调查了半叶眼眼对月球的代数性质。这些是作为月球基本杀菌者的代数,受关联性轴单体-单体轴轴轴轴线定义的埃伦堡-Moore代数值的分解法定义的分数。我们证明,如果基本类别有共产物,那么月球M的半代数实际上就是作为配给性法律的Eilenberg-Moore代数,以适合的月球结构,我们称之为半无月球M ⁇ 。我们还为可能月球、半组月球和有限分布元的半代数的半代数提供了具体的代数演示。第二项贡献将M T $\ rightroory$ T M 表格的薄弱分配法定性为强有力的分配性法律,但需附加附加条件。