Łukasiewicz logic and Riesz spaces
Łukasiewicz logic and Riesz spaces
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Łukasiewicz 逻辑和 Riesz 空间
DOI：
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发表时间：
2014
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通讯作者：
Ioana Leustean
中科院分区：
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作者：
A. Nola;Ioana Leustean
We initiate a deep study of Riesz MV-algebras which are MV-algebras endowed with a scalar multiplication with scalars from $$[0,1]$$[0,1]. Extending Mundici’s equivalence between MV-algebras and $$ell $$ℓ-groups, we prove that Riesz MV-algebras are categorically equivalent to unit intervals in Riesz spaces with strong unit. Moreover, the subclass of norm-complete Riesz MV-algebras is equivalent to the class of commutative unital C$$^*$$∗-algebras. The propositional calculus $${mathbb R}{mathcal L}$$RL that has Riesz MV-algebras as models is a conservative extension of Łukasiewicz $$infty $$∞-valued propositional calculus and is complete with respect to evaluations in the standard model $$[0,1]$$[0,1]. We prove a normal form theorem for this logic, extending McNaughton theorem for Ł ukasiewicz logic. We define the notions of quasi-linear combination and quasi-linear span for formulas in $${mathbb R}{mathcal L},$$RL, and relate them with the analogue of de Finetti’s coherence criterion for $${mathbb R}{mathcal L}$$RL.
