Łukasiewicz–Moisil algebra

A LM_n algebra is a De Morgan algebra (a notion also introduced by Moisil) with n-1 additional unary, "modal" operations: $\backslash nabla\_1,\; \backslash ldots,\; \backslash nabla\_\{n-1\}$, i.e. an algebra of signature $(A,\; \backslash vee,\; \backslash wedge,\; \backslash neg,\; \backslash nabla\_\{j\; \backslash in\; J\},\; 0,\; 1)$ where J = { 1, 2, ... n-1 }. (Some sources denote the additional operators as $\backslash nabla^n\_\{j\; \backslash in\; J\}$ 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. {{doi|10.1007/978-3-540-75939-3_5}}) The additional unary operators ∇_j must satisfy the following axioms for all x, y ∈ A and j, k ∈ J:

1. $\backslash nabla\_j(x\; \backslash vee\; y)\; =\; (\backslash nabla\_j\backslash ;\; x)\; \backslash vee\; (\backslash nabla\_j\backslash ;\; y)$
2. $\backslash nabla\_j\backslash ;x\; \backslash vee\; \backslash neg\; \backslash nabla\_j\backslash ;x\; =\; 1$
3. $\backslash nabla\_j\; (\backslash nabla\_k\backslash ;x)\; =\; \backslash nabla\_k\backslash ;x$
4. $\backslash nabla\_j\; \backslash neg\; x\; =\; \backslash neg\; \backslash nabla\_\{n-j\}\backslash ;x$
5. $\backslash nabla\_1\backslash ;x\backslash leq\; \backslash nabla\_2\backslash ;x\backslash cdots\backslash leq\; \backslash nabla\_\{n-1\}\backslash ;x$
6. if $\backslash nabla\_j\backslash ;x\; =\; \backslash nabla\_j\backslash ;y$ 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.)

The duals of some of the above axioms follow as properties:

• $\backslash nabla\_j(x\; \backslash wedge\; y)\; =\; (\backslash nabla\_j\backslash ;\; x)\; \backslash wedge\; (\backslash nabla\_j\backslash ;\; y)$
• $\backslash nabla\_j\backslash ;x\; \backslash wedge\; \backslash neg\; \backslash nabla\_j\backslash ;x\; =\; 0$

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

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

:$\backslash nabla\_j\backslash left(\backslash frac\{i\}\{n-1\}\backslash right)=\; \backslash ;\; \backslash begin\{cases\}$

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.

Moisil proved that every LM_n algebra can be embedded in a direct product (of copies) of the canonical $\backslash mathcal\{L\}\_n$ algebra. As a corollary, every LM_n algebra is a subdirect product of subalgebras of $\backslash mathcal\{L\}\_n$.

The Heyting implication can be defined as:

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

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, António [http://purl.pt/2926 "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."

{{reflist}}

• {{cite book|author1=Raymond Balbes|author2=Philip Dwinger|title=Distributive lattices|year=1975|publisher=University of Missouri Press|isbn=978-0-8262-0163-8|at=Chapter IX. De Morgan Algebras and Lukasiewicz Algebras}}
• 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

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Category:Algebraic logic

Category:Ockham algebras