## Tbilisi Mathematical Journal

### The field $\mathbb F_8$ as a Boolean manifold

René Guitart

#### Abstract

In a previous paper (“Hexagonal Logic of the Field $\mathbb F_8$ as a Boolean Logic with Three Involutive Modalities”, in The road to Universal Logic), we proved that elements of $\mathbb P(8)$, i.e. functions of all finite arities on the Galois field $\mathbb F_8$, are compositions of logical functions of a given Boolean structure, plus three geometrical cross product operations. Here we prove that $\mathbb P(8)$ admits a purely logical presentation, as a Boolean manifold, generated by a diagram of $4$ Boolean systems of logical operations on $\mathbb F_8$. In order to obtain this result we provide various systems of parameters of the set of unordered bases on $\mathbb F_2^3$, and consequently parametrical polynomial expressions for the corresponding conjunctions, which in fact are enough to characterize these unordered bases (and the corresponding Boolean structures).

#### Article information

Source
Tbilisi Math. J., Volume 8, Issue 1 (2015), 31-62.

Dates
Accepted: 21 October 2014
First available in Project Euclid: 12 June 2018

https://projecteuclid.org/euclid.tbilisi/1528768987

Digital Object Identifier
doi:10.1515/tmj-2015-0002

Mathematical Reviews number (MathSciNet)
MR3314180

Zentralblatt MATH identifier
1322.03020

#### Citation

Guitart, René. The field $\mathbb F_8$ as a Boolean manifold. Tbilisi Math. J. 8 (2015), no. 1, 31--62. doi:10.1515/tmj-2015-0002. https://projecteuclid.org/euclid.tbilisi/1528768987

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