Łukasiewicz–Moisil algebra


Łukasiewicz–Moisil algebras were introduced in the 1940s by Grigore Moisil 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 Łukasiewicz–Tarski logic was provided by C. C. Chang's MV-algebra, introduced in 1958. For the axiomatically more complicated 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, 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 in the framework of Brower’s intuitionistic logic.

Definition

A LMn algebra is a De Morgan algebra with n-1 additional unary, "modal" operations:, i.e. an algebra of signature where J =. The additional unary operators ∇j must satisfy the following axioms for all x, y ∈ A and j, k ∈ J:
  6. if for all j ∈ J, then x = y.

    Elementary properties

The duals of some of the above axioms follow as properties:
Additionally: and. In other words, the unary "modal" operations are lattice endomorphisms.

Examples

LM2 algebras are the Boolean algebras. The canonical Łukasiewicz algebra that Moisil had in mind were over the set L_n = with negation conjunction and disjunction and the unary "modal" operators:
If B is a Boolean algebra, then the algebra over the set B ≝ with the lattice operations defined pointwise and with ¬ ≝, and with the unary "modal" operators ∇2 ≝ and ∇1 = ¬∇2¬ = is a three-valued Łukasiewicz algebra.

Representation

Moisil proved that every LMn algebra can be embedded in a direct product of the canonical algebra. As a corollary, every LMn algebra is a subdirect product of subalgebras of.
The Heyting implication can be defined as:
Antonio Monteiro showed that for every monadic Boolean algebra one can construct a trivalent Łukasiewicz algebra 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."