# Algebraic Loops (2); Symmetry-Conserving Vector-Division Hoop Algebras

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An algebraic loop with the Moufang property, for all , , in the loop, acts as a vector multiplication and division table for unsigned vectors. If and , then ; the left-multiplicative inverse is *.*

Contributed by: Roger Beresford (March 2011)

Open content licensed under CC BY-NC-SA

## Snapshots

## Details

Algebraic loop Demonstrations are based on the GroupLoopHoop package in *MathSource*/4894, which contains much more information. The first develops loop properties and the third describes small groups and Moufang loops. Demonstrations of Abelian loops with Cartesian/polar duals, and Clifford (geometric) algebras (which are anticommutative hoops) are being prepared.

Integer elements are used for compactness, but real and complex numbers behave similarly. Output is real, to avoid bulky results involving square-root symbols.

Multiplication . If the table has in row and column , then the element of the product is the sum of A[[i]] B[[j]] Sign[k]. If the product has zero sizes where or has a nonzero size (because and have disparate zero sizes), nonzero remainders are created with the missing size(s). The left remainder is and the right remainder is . Compare this with integer division with remainder and , so recovers . As every vector has an inverse, division is simply multiplication by an inverse.

Inverse . If the shape of has symbolic form S[[k]] (a list of polynomials) and the table has a 1 (possibly signed) in row of column (this data is supplied in ), then the element of the (left) inverse is the signed sum of the derivations of S[[k] with respect to symbolic variable[[n]], divided by S[[k]]. This partial fraction is omitted if S[[k]] is zero; the zero has been factored out of the determinant.

An outstanding research problem is the development of compact *shape* expressions for some tables (some are shown with "{0}" in their shapes); division cannot be implemented in such cases.

The discovery that hoops (including all Clifford algebras) have exactly conserved sizes (and not just norms with triangle inequalities) appears to be new.

Hoops include compact descriptions of all the polar-complex and planar-complex analytic algebras developed (at great length, but without *shapes*) in Sylviu Olariu's "Complex Numbers in n Dimensions".

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