# Algebraic Loops (1); Properties

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An algebraic loop (or reduced quasigroup) describes the closed binary multiplication of a set of "unsigned elements" (the product of and appears in the position). The companion Demonstration, Algebraic Loops (2), shows how an algebraic loop also acts as a Cayley multiplication table for unsigned vectors (ordered sets of elements such as ).

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Contributed by: Roger Beresford (March 2011)

Open content licensed under CC BY-NC-SA

## Snapshots

## Details

This Demonstration only handles simple tests on small finite loops. The Demonstrations Algebraic Loops (2) and (3) investigate the algebraic properties of loops, dealing with symmetry-conserving partial-fraction vector-division "hoop algebras" with generalized signs. Hoops (which, by definition, conserve their determinants as symmetries) have the Moufang properties, ensuring that every vector has a multiplicative inverse. Later Demonstrations are concerned with the non-algebraic properties of associative and alternative tables (groups and octonions).

The GroupLoopHoop7.m package ([1], in *MathSource*/4894) is the source of the data, procedures, and nomenclature. and are the cyclic and quaternionic groups with elements. is the dihedral group with elements; has 12 elements; implies creation from Pauli -matrices; implies a non-conservative loop; implies a non-associative Moufang loop; etc. are direct compositions; trailing *, **,* etc indicates isomorphs.

The main reference for loops is

[2] J. D. H. Smith and A. B. Romanowska, Post-Modern Algebra, New York: Wiley Interscience, 1999.

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