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: $\nabla _{1},\ldots ,\nabla _{n-1}$ , 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.[7]) The additional unary operators ∇_j must satisfy the following axioms for all x, y ∈ A and j, k ∈ J:[3]

$\nabla _{j}(x\vee y)=(\nabla _{j}\;x)\vee (\nabla _{j}\;y)$
$\nabla _{j}\;x\vee \neg \nabla _{j}\;x=1$
$\nabla _{j}(\nabla _{k}\;x)=\nabla _{k}\;x$
$\nabla _{j}\neg x=\neg \nabla _{n-j}\;x$
$\nabla _{1}\;x\leq \nabla _{2}\;x\cdots \leq \nabla _{n-1}\;x$
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:[3]

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

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

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, 1/(n − 1), 2/(n − 1), ..., (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.[7]

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 ${\mathcal {L}}_{n}$ .[3]

The Heyting implication can be defined as:[6]

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

Andrei Popescu, Łukasiewicz-Moisil Relation Algebras, Studia Logica, Vol. 81, No. 2 (Nov., 2005), pp. 167-189
Lavinia Corina Ciungu (2013). Non-commutative Multiple-Valued Logic Algebras. Springer. pp. vii–viii. ISBN 978-3-319-01589-7.
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
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
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)
Georgescu, G. (2006). "N-Valued Logics and Łukasiewicz–Moisil Algebras". Axiomathes. 16: 123. doi:10.1007/s10516-005-4145-6., Theorem 3.6
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
Monteiro, António "Sur les algèbres de Heyting symétriques." Portugaliae mathematica 39.1–4 (1980): 1–237. Chapter 7. pp. 204-206