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 => T M as strong distributive laws M^s T => 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推动,我们调查了半叶眼眼对月球的代数性质。这些代数是月球根部的根菌的代数,受关联性轴单体-单位轴轴的制约。这些代数来自艾伦堡-摩尔代数系定义的下降。我们证明,如果基本类别有共产物,那么月球M的半叶眼镜事实上就是爱伦堡-摩尔代数系适合的月经结构。我们称之为半无月经M ⁇ 。我们还为可能月经、半组月经和有限分布元甲的半代数提供了具体的半代数表。第二个贡献是将表M T ⁇ T M 的薄弱分配法定性为强有力的分配法 MQ ⁇ T ⁇ T M ⁇ M 作附加条件。