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Hexagonal Logic of the Field \(\mathbb{F}_{8}\) as a Boolean Logic with Three Involutive Modalities

  • René Guitart
Part of the Studies in Universal Logic book series (SUL)

Abstract

We consider the Post–Malcev full iterative algebra \(\mathbb{P}_{8}\) of all functions of all finite arities on a set 8 with 8 elements, e.g. on the Galois field \(\mathbb{F}_{8}\). We prove that \(\mathbb{P}_{8}\) is generated by the logical operations of a canonical boolean structure on \(\mathbb{F}_{8} = \mathbb{F}_{2}^{3}\), plus three involutive \(\mathbb{F}_{2}\)-linear transvections A,B,C, related by circular relations and generating the group \(\operatorname {GL}_{3}(\mathbb{F}_{2})\). It is known that \(\operatorname {GL}_{3}(\mathbb{F}_{2}) = \operatorname {PSL}_{2}(\mathbb{F}_{7}) = \operatorname {G}_{168}\) is the unique simple group of order 168, which is the group of automorphisms of the Fano plane. Also we obtain that \(\mathbb{P}_{8}\) is generated by its boolean logic plus the three cross product operations R×, S×, I×.

Especially, our result could be understood as a hexagonal logic, a natural setting to study the logic of functions on a hexagon; precisely, it is a hexagonal presentation of the logic of functions on a cube with a selected diagonal.

Keywords

Hexagon of opposition Borromean object Specular logic Boolean algebra Modality Many-valued logics Finite fields Fano plane 

Mathematics Subject Classification (2000)

00A06 03B45 03B50 03G05 06Exx 06E25 06E30 11Txx 

References

  1. 1.
    Béziau, J.-Y.: The power of the hexagon. Log. Univers. 6(1–2), 1–43 (2012) MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Blanché, R.: Sur l’opposition des concepts. Theoria 19 (1953) Google Scholar
  3. 3.
    Blanché, R.: Structures Intellectuelles. Vrin, Paris (1966) Google Scholar
  4. 4.
    Dickson, L.E.: Linear Groups. Dover, New York (1958) (1st ed. 1900) zbMATHGoogle Scholar
  5. 5.
    Guitart, R.: L’idée de logique spéculaire. In: Journées Catégories, Algèbres, Esquisses, Néo-esquisses, Caen, 27–30 September 1994 Google Scholar
  6. 6.
    Guitart, R.: Moving logic, from Boole to Galois. In: Colloque International “Charles Ehresmann: 100 ans”, 7–9 October 2005, Amiens. Cahiers Top Géo Diff Cat, vol. XLVI-3, pp. 196–198 (2005) Google Scholar
  7. 7.
    Guitart, R.: Klein’s group as a Borromean object. In: Cahiers Top. Géo. Diff. Cat, vol. L-2, pp. 144–155 (2009) Google Scholar
  8. 8.
    Guitart, R.: A hexagonal framework of the field \(\mathbb{F}_{4}\) and the associated Borromean logic. Log. Univers. 6(1–2), 119–147 (2012) MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Heisler, J.: A characterization of finite fields. Am. Math. Mon. 74, 537–538, 1211 (1967) Google Scholar
  10. 10.
    Lau, D.: Function Algebras on Finite Sets. Springer, Berlin (2006) zbMATHGoogle Scholar
  11. 11.
    Lenstra, H.W. Jr., Schoof, R.J.: Primitive normal basis for finite fields. Math. Comput. 48, 217–231 (1987) MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Lidl, R., Niederreiter, H.: Introduction to Finite Fields and Their Applications. C.U.P., Cambridge (1994) CrossRefzbMATHGoogle Scholar
  13. 13.
    Malcev, A.I.: Iterative algebra and Post’s varieties. Algebra Log. 5, 5–24 (1966) (Russian) MathSciNetGoogle Scholar
  14. 14.
    Moore, E.H.: Mathematical Papers, Chicago Congress of 1893, pp. 208–242; Bull. Am. Math. Soc., December 1893 Google Scholar
  15. 15.
    Ribenboim, P.: L’arithmétique des Corps. Hermann, Paris (1972) zbMATHGoogle Scholar
  16. 16.
    Sesmat, A.: Logique: 1. les Définitions, les Jugements. 2. Les Raisonnements, la Logistique. Hermann, Paris (1951) Google Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.IMJ-PRG Université Paris DiderotParisFrance

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