Łukasiewicz–Moisil algebra

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: $$\nabla_1, \ldots, \nabla_{n-1}$$, i.e. an algebra of signature $$(A, \vee, \wedge, \neg, \nabla_{j \in J}, 0, 1)$$ where J = { 1, 2, ... n-1 }. (Some sources denote the additional operators as $$\nabla^n_{j \in J}$$ 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) $$\nabla_j(x \vee y) = (\nabla_j\; x) \vee (\nabla_j\; y)$$
 * 2) $$\nabla_j\;x \vee \neg \nabla_j\;x = 1$$
 * 3) $$\nabla_j (\nabla_k\;x) = \nabla_k\;x$$
 * 4) $$\nabla_j \neg x = \neg \nabla_{n-j}\;x $$
 * 5) $$\nabla_1\;x\leq \nabla_2\;x\cdots\leq \nabla_{n-1}\;x$$
 * 6) if $$\nabla_j\;x = \nabla_j\;y$$ 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:
 * $$\nabla_j(x \wedge y) = (\nabla_j\; x) \wedge (\nabla_j\; y)$$
 * $$\nabla_j\;x \wedge \neg \nabla_j\;x = 0$$

Additionally: $$\nabla_j\;0 =0$$ and $$\nabla_j\;1 =1$$. In other words, the unary "modal" operations $$\nabla_j$$ are lattice endomorphisms.

Examples
LM2 algebras are the Boolean algebras. The canonical Łukasiewicz algebra $$\mathcal{L}_n$$ that Moisil had in mind were over the set $$ L_{n} = \{ 0,\ \frac{1} {n-1}, \frac{2} {n-1}, ... , \frac{n-2} {n-1}\, 1 \} $$ with negation $$\neg x = 1-x$$ conjunction $$x \wedge y = \min\{x, y \}$$ and disjunction $$x \vee y = \max\{x, y \}$$ and the unary "modal" operators:
 * $$\nabla_j\left(\frac{i}{n-1}\right)= \; \begin{cases}

0 & \mbox{if } i+j < n \\ 1 & \mbox{if } i+j \geq n \\ \end{cases} \quad i \in \{0\} \cup J,\; j \in 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 $$\mathcal{L}_n$$ algebra. As a corollary, every LMn algebra is a subdirect product of subalgebras of $$\mathcal{L}_n$$.

The Heyting implication can be defined as:
 * $$x \Rightarrow y\; \overset{\mathrm{def}}{=}\;y \vee \bigwedge_{j\in J}(\neg\nabla_j\;x) \vee (\nabla_j\;y)$$

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."