What the łukasiewicz axioms mean

Journal of Symbolic Logic 85 (3):906-917 (2020)
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Abstract

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: $x\to = y\to $ and $x\to y=1 $ iff $x\leq y$. Let $\neg x=x\to 0$. Then the algebra $W=$ satisfies the time-honored Łukasiewicz axioms of his infinite-valued calculus. Let $x\to _{\text {\tiny \L }}y=\min $ 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 }}= $. Thus $x\to y= \phi ^{-1}+\phi ))$.

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AF-algebras with lattice-ordered K0: Logic and computation.Daniele Mundici - 2023 - Annals of Pure and Applied Logic 174 (1):103182.

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