## Tbilisi Mathematical Journal

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

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

#### 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**

Received: 21 October 2014

Accepted: 21 October 2014

First available in Project Euclid: 12 June 2018

**Permanent link to this document**

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

**Subjects**

Primary: 03B50: Many-valued logic

Secondary: 03G05: Boolean algebras [See also 06Exx] 06Exx: Boolean algebras (Boolean rings) [See also 03G05] 06E25: Boolean algebras with additional operations (diagonalizable algebras, etc.) [See also 03G25, 03F45] 06E30: Boolean functions [See also 94C10] 11Txx: Finite fields and commutative rings (number-theoretic aspects)

**Keywords**

Boolean algebra many-valued logics finite fields

#### 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