A logical approach to Riesz spaces

MV-modules were defined in [1, 2] as MV-algebras endowed with a structure of module over a PMV-algebra. When the PMV-algebra is the standard MV-algebra with the real product operation ([0, 1], ·), the corresponding MV-modules will be called Riesz MV-algebras. The class of Riesz MV-algebras is categorically equivalent (through a Mundici type theorem) with a class of Riesz spaces with strong unit. We shall present a propositional calculus for Riesz MV-algebras, denoted RL. If L is the propositional calculus of ÃLukasiewicz logic then: (1) the language of RL is the language of L, enriched with a set of unary logical connectives {4a: for any a ∈ [0, 1]}, (2) the axioms of RL are: I. the axioms of L, II. the following formulas are axioms (where a, b ∈ A): (R1) 4a(ϕ → ψ) → (4aϕ→4aψ), (R2) (4aϕ→4aψ)→4a(ϕ → ψ)