In a previous paper ('Hexagonal Logic of the Field F₈ as a Boolean Logic with Three Involutive Modalities', in 'The road to Universal Logic'), we proved that elements of P(8), i.e. functions of all finite arities on the Galois field F₈, are compositions of logical functions of a given Boolean structure, plus three geometrical cross product operations. Here we prove that P(8) admits a purely logical presentation, as a Boolean manifold, generated by a diagram of 4 Boolean systems of logical operations on F₈. In order to obtain this result we provide various systems of parameters of the set of unordered bases on F₂³, and consequently parametrical polynomial expressions for the corresponding conjunctions, which in fact are enough to characterize these unordered bases (and the corresponding Boolean structures).

Tbilisi Mathematical Journal, Vol. 8(1) (2015), pp. 31-62