Łukasiewicz–Moisil algebra

(Redirected from Łukasiewicz algebra)

Łukasiewicz–Moisil algebras (LMn 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 MVn-algebras.[2] MVn-algebras are a subclass of LMn-algebras, and the inclusion is strict for n ≥ 5.[3] In 1982 Roberto Cignoli published some additional constraints that added to LMn-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 LMθ algebras.[5] Although the Łukasiewicz implication cannot be defined in a LMn algebra for n ≥ 5, the Heyting implication can be, i.e. LMn 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

edit

A LMn algebra is a De Morgan algebra (a notion also introduced by Moisil) with n-1 additional unary, "modal" operations:  , i.e. an algebra of signature   where J = { 1, 2, ... n-1 }. (Some sources denote the additional operators as   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, yA and j, kJ:[3]

  1.  
  2.  
  3.  
  4.  
  5.  
  6. if   for all jJ, 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

edit

The duals of some of the above axioms follow as properties:[3]

  •  
  •  

Additionally:   and  .[3] In other words, the unary "modal" operations   are lattice endomorphisms.[6]

Examples

edit

LM2 algebras are the Boolean algebras. The canonical Łukasiewicz algebra   that Moisil had in mind were over the set   with negation   conjunction   and disjunction   and the unary "modal" operators:

 

If B is a Boolean algebra, then the algebra over the set B[2] ≝ {(x, y) ∈ B×B | xy} 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

edit

Moisil proved that every LMn algebra can be embedded in a direct product (of copies) of the canonical   algebra. As a corollary, every LMn algebra is a subdirect product of subalgebras of  .[3]

The Heyting implication can be defined as:[6]

 

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

edit
  1. ^ Andrei Popescu, Łukasiewicz-Moisil Relation Algebras, Studia Logica, Vol. 81, No. 2 (Nov., 2005), pp. 167-189
  2. ^ a b Lavinia Corina Ciungu (2013). Non-commutative Multiple-Valued Logic Algebras. Springer. pp. vii–viii. ISBN 978-3-319-01589-7.
  3. ^ a b c d e Iorgulescu, A.: Connections between MVn-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. ^ a b c 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. ^ a b c d 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

edit
  • 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 MVn-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 MVn-algebras and n-valued Łukasiewicz-Moisil—III. Unpublished Manuscript
  • Iorgulescu, A.: Connections between MVn-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