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

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


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.


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 


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Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.IMJ-PRG Université Paris DiderotParisFrance

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