Tbilisi Mathematical Journal

The field $\mathbb F_8$ as a Boolean manifold

René Guitart

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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


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