User:Junewe/Bertolino algebra

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Bertolino algebra is a subarea of abstract algebra which expands a field with n-ary multiple inverses of a scalar element. Unlike ordinary n-space in a field where the binary operations addition and subraction are used to identify the null-element, bertolino algebra has no restriction of such. In bertolino algebra it is possible to write the null-element zero as a combination of n-ary operations on n-coloured scalar elements.

Terminology

Colour

The colour of an element is what describes the scalar elements sign. In ordinary algebra + and - are used to denote the sign of the scalar element as an example. Colour could be interpretated as a broader meaning of the sign of a scalar.

Example of bertolino algebra

Total dependence of inverses

Let + be a binary operation on the operant a.

λ = (Θa) + (+a) + (-a) = 0, for every element a, a has a bivalent additive inverse. All (three) colours are needed to sum up to zero. We say that a is bivalently dependent to inverses and total dependent to inverses.

Non-total dependence of inverses

Let + be an binary operation on the operant a.

λ = (Θa) + (+a) = (Θa) + (-a) = (+a) + (-a) = 0, for every element a, a has two additive inverses. In this example, only two out of three colours are needed to sum up to zero. We say that (+a) is non-total dependent since only one other colour is needed to sum up to zero.

Opposite dependence of inverses

Let Θ, -, + be terniary operations on a.

λ = (+a) Θ (-a) = (Θa) - (+a) = (Θa) + (-a) = 0 Each of the three colours of a (+a), (-a), (Θa) has an inverse opposite to the terniary operation +, -, Θ. In this example, we say that (+a) is opposite dependent to (-a) as the terniary operation Θ is required to sum up to zero.