Sédénion

In Mathematical, the sédénions , noted \ mathbb S, form a Algèbre with 16 Dimension S on the Réel S. Their name comes from Latin sedecim who wants to say sixteen. Two kinds are currently known: 1) Sédénions obtained by application of the construction of Cayley-Dickson and the 2) sédénions conical (or algebra M) from the arithmetic ones of the hypernombres.

Sédénions of the construction of Cayley-Dickson

Arithmetic

Following the example octonions them, the Multiplication sedénions is neither commutative nor associative. Moreover, compared to octonions, sédénions them lose the property to be alternate.

Sédénions have a neutral element multiplicative 1 and opposite for the multiplication, but they do not form an algebra of division. That because they have dividing of zero.

Each sedénion is a linear combination, with real coefficients, sédénions units 1, E 1, E 2, E 3, E 4, E 5, E 6, E 7, E 8, E 9, E 10, E 11, E 12, E 13, E 14 and E 15, which forms the base of the vector Space of sédénions. The Multiplication table of these sédénions unit is established as follows:

The sédénions conical/algebra M with 16-dim.

Arithmetic

With the difference of sédénions resulting from the construction of Cayley-Dickson, which is built on the unit (1) and 15 root of the negative unit (- 1), sédénions them conical are built on 8 square roots of the positive and negative unit. They divide to it not Commutativité and it not Associativité with the arithmetic one of sédénions of Cayley-Dickson (" sédénions circulaire"), nevertheless sédénions them conical are modular, alternate, flexible but powers are not associative.

Sédénions conical contain at the same time the subalgebras of the Octonion S (circulars), the octonions conical and the octonions hyperbolic. Octonions hyperbolic are in a calculative way equivalents to the octonions split.

Sédénions conical contain elements Idempotent S, Nilpotent S and dividing of zero. With the exception from their elements nilpotents and zero, the arithmetic one is closed with the respect of the operations of power and logarithm.

Dependant subjects

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