Łukasiewicz–Moisil algebra

Łukasiewicz–Moisil algebras (LM_n algebras) were introduced in the 1940s by Grigore Moisil (initially under the name of Łukasiewicz algebras^[1]) 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.^[2] MV_n-algebras are a subclass of LM_n-algebras, and the inclusion is strict for n ≥ 5.^[3] 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.^[4]

Moisil however, published in 1964 a logic to match his algebra (in the general n ≥ 5 case), now called Moisil logic.^[2] 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.^[5] 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.^[6]

Definition

A LM_n algebra is a De Morgan algebra (a notion also introduced by Moisil) with n-1 additional unary, "modal" operations: ${\displaystyle \nabla _{1},\ldots ,\nabla _{n-1}}$, i.e. an algebra of signature ${\displaystyle (A,\vee ,\wedge ,\neg ,\nabla _{j\in J},0,1)}$ where J = { 1, 2, ... n-1 }. (Some sources denote the additional operators as ${\displaystyle \nabla _{j\in J}^{n}}$ to emphasize that they depend on the order n of the algebra.^[7]) The additional unary operators ∇_j must satisfy the following axioms for all x, y ∈ A and j, k ∈ J:^[3]

1. ${\displaystyle \nabla _{j}(x\vee y)=(\nabla _{j}\;x)\vee (\nabla _{j}\;y)}$
2. ${\displaystyle \nabla _{j}\;x\vee \neg \nabla _{j}\;x=1}$
3. ${\displaystyle \nabla _{j}(\nabla _{k}\;x)=\nabla _{k}\;x}$
4. ${\displaystyle \nabla _{j}\neg x=\neg \nabla _{n-j}\;x}$
5. ${\displaystyle \nabla _{1}\;x\leq \nabla _{2}\;x\cdots \leq \nabla _{n-1}\;x}$
6. if ${\displaystyle \nabla _{j}\;x=\nabla _{j}\;y}$ 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:^[3]

• ${\displaystyle \nabla _{j}(x\wedge y)=(\nabla _{j}\;x)\wedge (\nabla _{j}\;y)}$
• ${\displaystyle \nabla _{j}\;x\wedge \neg \nabla _{j}\;x=0}$

Additionally: ${\displaystyle \nabla _{j}\;0=0}$ and ${\displaystyle \nabla _{j}\;1=1}$.^[3] In other words, the unary "modal" operations ${\displaystyle \nabla _{j}}$ are lattice endomorphisms.^[6]

Examples

LM₂ algebras are the Boolean algebras. The canonical Łukasiewicz algebra ${\displaystyle {\mathcal {L}}_{n}}$ that Moisil had in mind were over the set ${\displaystyle L_{n}=\{0,\ {\frac {1}{n-1}},{\frac {2}{n-1}},...,{\frac {n-2}{n-1}}\ ,1\}}$ with negation ${\displaystyle \neg x=1-x}$ conjunction ${\displaystyle x\wedge y=\min\{x,y\}}$ and disjunction ${\displaystyle x\vee y=\max\{x,y\}}$ and the unary "modal" operators:

${\displaystyle \nabla _{j}\left({\frac {i}{n-1}}\right)=\;{\begin{cases}0&{\mbox{if }}i+j<n\\1&{\mbox{if }}i+j\geq n\\\end{cases}}\quad i\in \{0\}\cup J,\;j\in J.}$

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.^[7]

Representation

Moisil proved that every LM_n algebra can be embedded in a direct product (of copies) of the canonical ${\displaystyle {\mathcal {L}}_{n}}$ algebra. As a corollary, every LM_n algebra is a subdirect product of subalgebras of ${\displaystyle {\mathcal {L}}_{n}}$.^[3]

The Heyting implication can be defined as:^[6]

${\displaystyle x\Rightarrow y\;{\overset {\mathrm {def} }{=}}\;y\vee \bigwedge _{j\in J}(\neg \nabla _{j}\;x)\vee (\nabla _{j}\;y)}$

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.^[7][8] 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."^[7]

References

1. Andrei Popescu, Łukasiewicz-Moisil Relation Algebras, Studia Logica, Vol. 81, No. 2 (Nov., 2005), pp. 167-189
2. Lavinia Corina Ciungu (2013). Non-commutative Multiple-Valued Logic Algebras. Springer. pp. vii–viii. ISBN 978-3-319-01589-7.
3. Iorgulescu, A.: Connections between MV_n-algebras and n-valued Łukasiewicz-Moisil algebras—I. Discrete Math. 181, 155–177 (1998) doi:10.1016/S0012-365X(97)00052-6
4. R. Cignoli, Proper n-Valued Łukasiewicz Algebras as S-Algebras of Łukasiewicz n-Valued Propositional Calculi, Studia Logica, 41, 1982, 3–16, doi:10.1007/BF00373490
5. Georgescu, G., Iourgulescu, A., Rudeanu, S.: "Grigore C. Moisil (1906–1973) and his School in Algebraic Logic." International Journal of Computers, Communications & Control 1, 81–99 (2006)
6. Georgescu, G. (2006). "N-Valued Logics and Łukasiewicz–Moisil Algebras". Axiomathes. 16 (1–2): 123–136. doi:10.1007/s10516-005-4145-6., Theorem 3.6
7. Cignoli, R., "The algebras of Lukasiewicz many-valued logic - A historical overview," in S. Aguzzoli et al.(Eds.), Algebraic and Proof-theoretic Aspects of Non-classical Logics, LNAI 4460, Springer, 2007, 69-83. doi:10.1007/978-3-540-75939-3_5
8. Monteiro, António "Sur les algèbres de Heyting symétriques." Portugaliae Mathematica 39.1–4 (1980): 1–237. Chapter 7. pp. 204-206

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