## 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)

### Keywords:

Boolean algebra; many-valued logics; finite fields

Zbl 1352.03030
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### References:

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