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 ).[more]
The tables have indices in the first row and first column (making them "reduced"), and each index occurs once in each row and column (making them "quasigroups"). Many loops exist, in many isomorphic forms; a selection (with up to 16) demonstrates various features.
Algebraic loops with the associative property are groups, but other properties (Abelian, alternative, ..., Moufang, ..., etc.) are also important. 48 loops and 14 properties are offered here, together with three "folding" operations that convert symmetrical loops to algebras with "generalized signs". Signs , , and are taken to be fundamental concepts in most mathematical texts and are assumed to be unique square roots and fourth roots of +1. They arise from the folding of symmetrical multiplication tables. Their uniqueness is disproved by Clifford algebras , which involve entities that are square roots and entities that are fourth roots of +1, together with a "pseudoscalar" that is an anticommutative version of . Ternary symmetry leads to signs 𝕛 and 𝕁 that are cube roots of +1: and , , , . Fifth and higher prime roots are generators of groups, but are probably of little physical significance. The generalization of signs by the folding of loops appears to be a new concept.
Choose a loop and a test. The table, and the condition that is tested (for all , , … in the symbolic table) will appear, together with the result. Alternative loops (which include octonions) have "square-associativity" and subsume both associative loops (groups) and Moufang loops. The Moufang property ensures that any vector has a multiplicative inverse. Many more properties exist than are demonstrated here but most are of only specialized interest.
You can also test whether a table has -fold symmetry (for ). If it does (usually indicated in the descriptive header), the top-left corner of the table can be converted into an "-signed algebra" for "-signed vectors"; give algebras over real and complex fields with signs ; gives a complete "terplex" algebra with signs (primitive cube roots of +1, NOT cyclotomic complex numbers). If the loop can be folded, partitions will appear and indices greater than will be replaced by signed indices. Note that different isomorphisms of the same loop may not fold or may fold to different algebras.
Other Demonstrations develop algebras and generalized signs in more detail. I use the neologism "hoops" for "symmetry conserving vector-division algebras" (all groups, a few non-associative Moufang loops, and many signed-table algebras); the factors of their symbolic determinants are conserved symmetries that may (by Noether's theorem) relate to forces and particles. Symmetry conservation is central to modern physics, so non-conservative algebras are unlikely to be relevant to physics.[less]
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 (, 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
 J. D. H. Smith and A. B. Romanowska, Post-Modern Algebra, New York: Wiley Interscience, 1999.