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.Iorgulescu, A.: Connections between MV_''n''-algebras and ''n''-valued Łukasiewicz-Moisil algebras—I. Discrete Math. 181, 155–177 (1998) 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., Theorem 3.6

Definition

A LM_''n'' algebra is a De Morgan algebra (a notion also introduced by Moisil) with ''n''-1 additional unary, "modal" operations: $\nabla_1, \ldots, \nabla_$, i.e. an algebra of signature $(A, \vee, \wedge, \neg, \nabla_, 0, 1)$ where ''J'' = . (Some sources denote the additional operators as $\nabla^n_$ to emphasize that they depend on the order ''n'' of the algebra.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. ) The additional unary operators ∇_''j'' must satisfy the following axioms for all ''x'', ''y'' ∈ ''A'' and ''j'', ''k'' ∈ ''J'':

# $\nabla_j(x \vee y) = (\nabla_j\; x) \vee (\nabla_j\; y)$
# $\nabla_j\;x \vee \neg \nabla_j\;x = 1$
# $\nabla_j (\nabla_k\;x) = \nabla_k\;x$
# $\nabla_j \neg x = \neg \nabla_\;x$
# $\nabla_1\;x\leq \nabla_2\;x\cdots\leq \nabla_\;x$
# if $\nabla_j\;x = \nabla_j\;y$ for all ''j'' ∈ ''J'', then ''x'' = ''y''.

(The adjective "modal" is related to the ltimately failedprogram 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 $\mathcal_n$ that Moisil had in mind were over the set ''L_n'' = with negation $\neg x = 1-x$ conjunction $x \wedge y = \min\$ and disjunction $x \vee y = \max\$ and the unary "modal" operators:
:$\nabla_j\left(\frac\right)= \; \begin 0 & \mbox i+j < n \\ 1 & \mbox i+j \geq n \\ \end \quad i \in \ \cup J,\; j \in J.$

If ''B'' is a Boolean algebra, then the algebra over the set ''B'' ^{/sup> ≝ 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'') erived by axiom 4is 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 $\mathcal_n$ algebra. As a corollary, every LM_''n'' algebra is a subdirect product of subalgebras of $\mathcal_n$.

The Heyting implication can be defined as:
:$x \Rightarrow y\; \overset\;y \vee \bigwedge_(\neg\nabla_j\;x) \vee (\nabla_j\;y)$

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.Monteiro, Antóni
"Sur les algèbres de Heyting symétriques."
Portugaliae mathematica 39.1–4 (1980): 1–237. Chapter 7. pp. 204-206 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

Further reading

*
* Boicescu, V., Filipoiu, A., Georgescu, G., Rudeanu, S.: Łukasiewicz-Moisil Algebras. North-Holland, Amsterdam (1991)
* Iorgulescu, A.: Connections between MV_''n''-algebras and ''n''-valued Łukasiewicz–Moisil algebras—II. Discrete Math. 202, 113–134 (1999)
* 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)
* 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

{{DEFAULTSORT:Lukasiewicz-Moisil algebra
Algebraic logic
Ockham algebras