论文标题
各种有序代数的分类观点
A categorical view of varieties of ordered algebras
论文作者
论文摘要
众所周知，$σ$ -Algebras的经典品种与$ \ Mathsf {set} $上的Finarity Monad对应。我们为有序的$σ$ - 代数，即以$σ$ terms之间的不矛盾提出的类别提出了类似的结果。我们证明它们与$ \ mathsf {pos} $上的强烈限制的单子相对应。也就是说，那些保留反身涂层的限制单子。我们推断出，强烈的限制单子具有共同的陈述，类似于由于凯利和权力而引起的限制单子弹的呈现。我们还表明，这些单子是$ \ mathsf {set} $上的finalitary monad的举起。
It is well known that classical varieties of $Σ$-algebras correspond bijectively to finitary monads on $\mathsf{Set}$. We present an analogous result for varieties of ordered $Σ$-algebras, i.e., classes presented by inequations between $Σ$-terms. We prove that they correspond bijectively to strongly finitary monads on $\mathsf{Pos}$. That is, those finitary monads which preserve reflexive coinserters. We deduce that strongly finitary monads have a coinserter presentation, analogous to the coequaliser presentation of finitary monads due to Kelly and Power. We also show that these monads are liftings of finitary monads on $\mathsf{Set}$.
