Łukasiewicz–Moisil algebras (LM"n" 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 MV"n"-algebras. MV"n"-algebras are a subclass of LM"n"-algebras, and the inclusion is strict for "n" ≥ 5. In 1982 Roberto Cignoli published some additional constraints that added to LM"n"-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 LM"θ" algebras. Although the Łukasiewicz implication cannot be defined in a LM"n" algebra for "n" ≥ 5, the Heyting implication can be, i.e. LM"n" 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 LM"n" algebra is a De Morgan algebra (a notion also introduced by Moisil) with "n"-1 additional unary, "modal" operations: formula_1, i.e. an algebra of signature formula_2 where "J" = { 1, 2, ... "n"-1 }. (Some sources denote the additional operators as formula_3 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":
Elementary properties.
The duals of some of the above axioms follow as properties:
Additionally: formula_12 and formula_13. In other words, the unary "modal" operations formula_14 are lattice endomorphisms.
Examples.
LM2 algebras are the Boolean algebras. The canonical Łukasiewicz algebra formula_15 that Moisil had in mind were over the set formula_16 with negation formula_17 conjunction formula_18 and disjunction formula_19 and the unary "modal" operators:
formula_20
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 LM"n" algebra can be embedded in a direct product (of copies) of the canonical formula_15 algebra. As a corollary, every LM"n" algebra is a subdirect product of subalgebras of formula_15.
The Heyting implication can be defined as:
formula_23
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.
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