Hexagonal logic of the field $mathbb{f}_{8}$ as a boolean logic with three involutive modalities

René Guitart

pp. 191-220

in: Arnold Koslow, Arthur Buchsbaum (eds), The road to universal logic I, Basel, Birkhäuser, 2015

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.