Let $\to $ be a continuous $\protect \operatorname {\mathrm {[0,1]}}$ -valued function defined on the unit square $\protect \operatorname {\mathrm {[0,1]}}^2$ , having the following properties: (i) $x\to (y\to z)= y\to (x\to z)$ and (ii) $x\to y=1 $ iff $x\leq y$ . Let $\neg x=x\to 0$ . Then the algebra $W=(\protect \operatorname {\mathrm {[0,1]}},1,\neg ,\to )$ satisfies the time-honored Łukasiewicz axioms of his infinite-valued calculus. Let $x\to _{\text {\tiny \L }}y=\min (1,1-x+y)$ and $\neg _{\text {\tiny \L }}x=x\to _{\text {\tiny \L }} 0 =1-x.$ Then there is precisely one isomorphism $\phi $ of W onto the standard Wajsberg algebra $W_{\text {\tiny \L }}= (\protect \operatorname {\mathrm {[0,1]}},1,\neg _{\text {\tiny \L }},\to _{\text {\tiny \L }})$ . Thus $x\to y= \phi ^{-1}(\min (1,1-\phi (x)+\phi (y)))$ .

中文翻译：

---

ŁUKASIEWICZ 公理的含义

让$\到$成为一个连续的$\protect \operatorname {\mathrm {[0,1]}}$单位平方上定义的值函数$\protect \operatorname {\mathrm {[0,1]}}^2$，具有以下性质：(i)$x\to (y\to z)= y\to (x\to z)$(ii)$x\to y=1 $当且当$x\leq y$. 让$\neg x=x\to 0$. 然后代数$W=(\protect \operatorname {\mathrm {[0,1]}},1,\neg ,\to )$满足他的无限值微积分的历史悠久的 Łukasiewicz 公理。让$x\to _{\text {\tiny \L }}y=\min (1,1-x+y)$和$\neg _{\text {\tiny \L }}x=x\to _{\text {\tiny \L }} 0 =1-x.$那么恰好有一个同构$\phi $的W到标准 Wajsberg 代数$W_{\text {\tiny \L }}= (\protect \operatorname {\mathrm {[0,1]}},1,\neg _{\text {\tiny \L }},\to _{ \text {\tiny \L }})$. 因此$x\to y= \phi ^{-1}(\min (1,1-\phi (x)+\phi (y)))$.