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
).
Contributed by: Roger Beresford (March 2011)
Open content licensed under CC BY-NC-SA
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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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