Łukasiewicz–Moisil algebra

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: $\nabla _{1},\ldots ,\nabla _{n-1}$, 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 $\nabla _{j\in J}^{n}$ 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:

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 LM_n algebra can be embedded in a direct product (of copies) of the canonical ${\mathcal {L}}_{n}$ algebra. As a corollary, every LM_n 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."

Further reading

• Raymond Balbes; Philip Dwinger (1975). Distributive lattices. University of Missouri Press. Chapter IX. De Morgan Algebras and Lukasiewicz Algebras. ISBN 978-0-8262-0163-8.
• Boicescu, V., Filipoiu, A., Georgescu, G., Rudeanu, S.: Łukasiewicz-Moisil Algebras. North-Holland, Amsterdam (1991) ISBN 0080867898
• Iorgulescu, A.: Connections between MV_n-algebras and n-valued Łukasiewicz–Moisil algebras—II. Discrete Math. 202, 113–134 (1999) doi:10.1016/S0012-365X(98)00289-1
• Iorgulescu, A.: Connections between MV_n-algebras and n-valued Łukasiewicz-Moisil—III. Unpublished Manuscript
• Iorgulescu, A.: Connections between MV_n-algebras and n-valued Łukasiewicz–Moisil algebras—IV. J. Univers. Comput. Sci. 6, 139–154 (2000) doi:10.3217/jucs-006-01-0139
• R. Cignoli, Algebras de Moisil de orden n, Ph.D. Thesis, Universidad National del Sur, Bahia Blanca, 1969
• http://projecteuclid.org/download/pdf_1/euclid.ndjfl/1093635424