61.2 Elements of Free Magma Rings

Elements of families in this category have trivial normalisation, i.e., efficient methods for \= and \<.

In order to treat elements of free magma rings uniformly, also without an external representation, the attributes CoefficientsAndMagmaElements (see CoefficientsAndMagmaElements) and ZeroCoefficient (see ZeroCoefficient) were introduced that allow one to ``take an element of an arbitrary magma ring into pieces''.

Conversely, for constructing magma ring elements from coefficients and magma elements, ElementOfMagmaRing (see ElementOfMagmaRing) can be used. (Of course one can also embed each magma element into the magma ring, see Natural Embeddings related to Magma Rings, and then form the linear combination, but many unnecessary intermediate elements are created this way.)

is a list that contains at the odd positions the magma elements, and at the even positions their coefficients in the element elm.

For an element elm of a magma ring (modulo relations) RM, ZeroCoefficient returns the zero element of the coefficient ring R.

ElementOfMagmaRing returns the element å_{i = 1}ⁿ c_i m_i^¢, where coeffs = [ c₁, c₂, ¼, c_n ] is a list of coefficients, mgmelms = [ m₁, m₂, ¼, m_n ] is a list of magma elements, and m_i^¢ is the image of m_i under an embedding of a magma containing m_i into a magma ring whose elements lie in the family Fam. zerocoeff must be the zero of the coefficient ring containing the c_i.

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