论文标题
自由代数的函子从山地的角度来看
Free-algebra functors from a coalgebraic perspective
论文作者
论文摘要
给定一个方程式的$σ$，每个套装变量的$ x $ aussosiates $f_σ$ aSsociates a $ x $ the $ x $f_f_σ（x）$ by $ x $。将\ emph {derivative} $σ的概念扩展为任意集的方程式$σ$，最初由Dent，Kearnes和Szendrei定义，我们表明$f_σ$在$σ\vdashσ'$（即$ n $）（即$ nive $ nive $ derive $ derive $ derive $ derive $ derive $ derive $ derive $ derive $ nivive n时）。如果$f_σ$弱保留内核对，则每个方程式$ p（x，x，y）= q（x，y，y，y）$产生$ s（x，y，y，z，u）$，使得$ p（x，x，y，y，z）= s（x，x，y，y，z，z）$ and $ q（x，x，x，x，x，x，x，x，y，z）= s（x，x，x，x（x，x，x，y y，x，y y，x，y y，y，y y，y，y y，y y，y y y y y y y y y y y y y y y y y y y y，x，y y）$。在这种情况下，N-渗透品种必须已列出，即Mal'cev。相反，如果$σ$定义了mal'cev品种，则$f_σ$弱保留内核对。作为一种工具，我们证明，当且仅当它们弱地保留EPIS的回调时，任意$ set- $ endofunctors $ f $弱保存内核对。
Given a set $Σ$ of equations, the free-algebra functor $F_Σ$ associates to each set $X$ of variables the free algebra $F_Σ(X)$ over $X$. Extending the notion of \emph{derivative} $Σ'$ for an arbitrary set $Σ$ of equations, originally defined by Dent, Kearnes, and Szendrei, we show that $F_Σ$ preserves preimages if and only if $Σ\vdash Σ'$, i.e. $Σ$ derives its derivative $Σ'$. If $F_Σ$ weakly preserves kernel pairs, then every equation $p(x,x,y)=q(x,y,y)$ gives rise to a term $s(x,y,z,u)$ such that $p(x,y,z)=s(x,y,z,z)$ and $q(x,y,z)=s(x,x,y,z)$. In this case n-permutable varieties must already be permutable, i.e. Mal'cev. Conversely, if $Σ$ defines a Mal'cev variety, then $F_Σ$ weakly preserves kernel pairs. As a tool, we prove that arbitrary $Set-$endofunctors $F$ weakly preserve kernel pairs if and only if they weakly preserve pullbacks of epis.
