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

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.