The field \(\mathbb F_{8}\) as a Boolean manifold. (English) Zbl 1322.03020

Summary: In [the author, in: The road to universal logic. Festschrift for 50th birthday of Jean-Yves Béziau. Volume I. Cham: Birkhäuser/Springer. 191–220 (2015; Zbl 1352.03030)], 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).


03B50 Many-valued logic
03G05 Logical aspects of Boolean algebras
06E25 Boolean algebras with additional operations (diagonalizable algebras, etc.)
06E30 Boolean functions
11T30 Structure theory for finite fields and commutative rings (number-theoretic aspects)


Zbl 1352.03030
Full Text: DOI


[1] [1] M. Andreatta, A. Ehresmann, R. Guitart and G. Mazzola, Towards a categorical theory of creativity for music, discourse, and cognition, in J. Yust, J. Wild, and J.A. Burgoyne (Eds.): MCM 2013, LNAI 7937, pp. 19-37, 2013, Springer. · Zbl 1270.00024
[2] R. Guitart, Moving logic, from Boole to Galois, Colloque International “Charles Ehresmann : 100 ans”, 7-9 october 2005, Amiens, Cahiers Top Géo Diff Cat. vol. XLVI-3, 2005, 196-198. · Zbl 1076.03046
[3] R. Guitart, Klein’s group as a borromean object, Cahiers Top. Géo. Diff. Cat. vol. L-2, 2009, 144-155. · Zbl 1186.18002
[4] R. Guitart, A Hexagonal framework of the field F4 and the associated Borromean logic, Log. Univers.,6 (1-2), 2012, 119-147. · Zbl 1280.03021
[5] R. Guitart, Hexagonal logic of the field F8 as a boolean logic with three involutive modalities, in A. Koslow, A. Buchsbaum (eds.), The Road to Universal Logic, Studies in Universal Logic, Birkhäuser, 2015. p. 191-220. · Zbl 1352.03030
[6] D. Lau, Function Algebras on finite sets, Springer, 2006. · Zbl 1105.08001
[7] H.W. Lenstra Jr. and R.J. Schoof, Primitive normal basis for finite fields, Math. Comp., 48, 1987, 217-231. · Zbl 0615.12023
[8] R. Lidl and H. Niederreiter, Introduction to finite fields and their applications, C.U.P., 1994. · Zbl 0820.11072
[9] A. I. Malcev, Iterative algebra and Post’s varieties (Russian), Algebra i Logika (Sem.) 5, 1966, 5-24.
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