Łukasiewicz–Moisil algebras (LMn algebras) were introduced in the 1940s by Grigore Moisil (initially under the name of Łukasiewicz algebras) in the hope of giving algebraic semantics for the n-valued Łukasiewicz logic. However, in 1956 Alan Rose discovered that for n ≥ 5, the Łukasiewicz–Moisil algebra does not model the Łukasiewicz logic. A faithful model for the ℵ0-valued (infinitely-many-valued) Łukasiewicz–Tarski logic was provided by C. C. Chang's MV-algebra, introduced in 1958. For the axiomatically more complicated (finite) n-valued Łukasiewicz logics, suitable algebras were published in 1977 by Revaz Grigolia and called MVn-algebras. MVn-algebras are a subclass of LMn-algebras, and the inclusion is strict for n ≥ 5. In 1982 Roberto Cignoli published some additional constraints that added to LMn-algebras produce proper models for n-valued Łukasiewicz logic; Cignoli called his discovery proper Łukasiewicz algebras. Moisil however, published in 1964 a logic to match his algebra (in the general n ≥ 5 case), now called Moisil logic. After coming in contact with Zadeh's fuzzy logic, in 1968 Moisil also introduced an infinitely-many-valued logic variant and its corresponding LMθ algebras. Although the Łukasiewicz implication cannot be defined in a LMn algebra for n ≥ 5, the Heyting implication can be, i.e. LMn algebras are Heyting algebras; as a result, Moisil logics can also be developed (from a purely logical standpoint) in the framework of Brower's intuitionistic logic.
Definition A LMn algebra is a De Morgan algebra (a notion also introduced by Moisil) with n-1 additional unary, "modal" operations: ∇ 1 , … , ∇ n − 1 {\displaystyle \nabla _{1},\ldots ,\nabla _{n-1}} , i.e. an algebra of signature ( A , ∨ , ∧ , ¬ , ∇ j ∈ J , 0 , 1 ) {\displaystyle (A,\vee ,\wedge ,\neg ,\nabla _{j\in J},0,1)} where J = { 1, 2, ... n-1 }. (Some sources denote the additional operators as ∇ j ∈ J n {\displaystyle \nabla _{j\in J}^{n}} to emphasize that they depend on the order n of the algebra.) The additional unary operators ∇j must satisfy the following axioms for all x, y ∈ A and j, k ∈ J:
∇ j ( x ∨ y ) = ( ∇ j x ) ∨ ( ∇ j y ) {\displaystyle \nabla _{j}(x\vee y)=(\nabla _{j}\;x)\vee (\nabla _{j}\;y)}
∇ j x ∨ ¬ ∇ j x = 1 {\displaystyle \nabla _{j}\;x\vee \neg \nabla _{j}\;x=1}
∇ j ( ∇ k x ) = ∇ k x {\displaystyle \nabla _{j}(\nabla _{k}\;x)=\nabla _{k}\;x}
∇ j ¬ x = ¬ ∇ n − j x {\displaystyle \nabla _{j}\neg x=\neg \nabla _{n-j}\;x}
∇ 1 x ≤ ∇ 2 x ⋯ ≤ ∇ n − 1 x {\displaystyle \nabla _{1}\;x\leq \nabla _{2}\;x\cdots \leq \nabla _{n-1}\;x}
if ∇ j x = ∇ j y {\displaystyle \nabla _{j}\;x=\nabla _{j}\;y} for all j ∈ J, then x = y. (The adjective "modal" is related to the [ultimately failed] program of Tarksi and Łukasiewicz to axiomatize modal logic using many-valued logic.)
Elementary properties The duals of some of the above axioms follow as properties:
∇ j ( x ∧ y ) = ( ∇ j x ) ∧ ( ∇ j y ) {\displaystyle \nabla _{j}(x\wedge y)=(\nabla _{j}\;x)\wedge (\nabla _{j}\;y)}
∇ j x ∧ ¬ ∇ j x = 0 {\displaystyle \nabla _{j}\;x\wedge \neg \nabla _{j}\;x=0}
Additionally: ∇ j 0 = 0 {\displaystyle \nabla _{j}\;0=0} and ∇ j 1 = 1 {\displaystyle \nabla _{j}\;1=1} . In other words, the unary "modal" operations ∇ j {\displaystyle \nabla _{j}} are lattice endomorphisms.
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