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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).

MSC:

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)

Citations:

Zbl 1352.03030
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References:

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