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Łukasiewicz–Moisil algebra

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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.

Contents

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 , i.e. an algebra of signature ( A , ∨ , ∧ , ¬ , ∇ j ∈ J , 0 , 1 ) where J = { 1, 2, ... n-1 }. (Some sources denote the additional operators as ∇ j ∈ 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:

  1. ∇ j ( x ∨ y ) = ( ∇ j x ) ∨ ( ∇ j y )
  2. ∇ j x ∨ ¬ ∇ j x = 1
  3. ∇ j ( ∇ k x ) = ∇ k x
  4. ∇ j ¬ x = ¬ ∇ n − j x
  5. ∇ 1 x ≤ ∇ 2 x ⋯ ≤ ∇ n − 1 x
  6. if ∇ j x = ∇ 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 )
  • ∇ j x ∧ ¬ ∇ j x = 0

    • Additionally: ∇ j 0 = 0 and ∇ j 1 = 1 . In other words, the unary "modal" operations ∇ j are lattice endomorphisms.

    Examples

    LM2 algebras are the Boolean algebras. The canonical Łukasiewicz algebra L n that Moisil had in mind were over the set L_n = { 0, 1/(n − 1), 2/(n − 1), ..., (n-2)/(n-1), 1 } with negation ¬ x = 1 − x conjunction x ∧ y = min { x , y } and disjunction x ∨ y = max { x , y } and the unary "modal" operators:

    ∇ j ( i n − 1 ) = { 0 if  i + j < n 1 if  i + j ≥ n i ∈ { 0 } ∪ J , j ∈ J .

    If B is a Boolean algebra, then the algebra over the set B[2] ≝ {(x, y) ∈ B×B | x ≤ y} with the lattice operations defined pointwise and with ¬(x, y) ≝ (¬y, ¬x), and with the unary "modal" operators ∇2(x, y) ≝ (y, y) and ∇1(x, y) = ¬∇2¬(x, y) = (x, x) [derived by axiom 4] is a three-valued Łukasiewicz algebra.

    Representation

    Moisil proved that every LMn algebra can be embedded in a direct product (of copies) of the canonical L n algebra. As a corollary, every LMn algebra is a subdirect product of subalgebras of L n .

    The Heyting implication can be defined as:

    x ⇒ y = d e f y ∨ ⋀ j ∈ J ( ¬ ∇ j x ) ∨ ( ∇ j y )

    Antonio Monteiro showed that for every monadic Boolean algebra one can construct a trivalent Łukasiewicz algebra (by taking certain equivalence classes) and that any trivalent Łukasiewicz algebra is isomorphic to a Łukasiewicz algebra thus derived from a monadic Boolean algebra. Cignoli summarizes the importance of this result as: "Since it was shown by Halmos that monadic Boolean algebras are the algebraic counterpart of classical first order monadic calculus, Monteiro considered that the representation of three-valued Łukasiewicz algebras into monadic Boolean algebras gives a proof of the consistency of Łukasiewicz three-valued logic relative to classical logic."

    References

    Łukasiewicz–Moisil algebra Wikipedia
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