Dloops

Dloops, or D-loops, are a class of algebraic structures studied in loop theory, a branch of nonassociative algebra. In this context, a loop is a set L with a binary operation such that there exists an identity element e in L with e*a = a and a*e = a for all a in L, and such that for any a, b in L there exist unique x, y in L with a*x = b and y*a = b. D-loops are defined by enforcing dihedral-like symmetry constraints on this basic framework, yielding finite loops whose multiplication is designed to resemble certain dihedral-group aspects while not requiring associativity.

Construction and definitions: Several common approaches exist to construct D-loops. One approach uses a base group and introduces a nontrivial factor set or cocycle that implements a reflection-like involution, producing a nonassociative product with dihedral flavor. Another method is a dihedral-type doubling or twisted product that preserves a dihedral pair of generators in spirit.

Properties and examples: D-loops are nonassociative in general but retain the loop property (existence of identity and division). They often have additional symmetry properties encoded by autotopisms related to dihedral actions. Small-order examples have been cataloged, and some D-loops are isotopic to dihedral groups or arise as deformations of them. Their multiplication tables exhibit patterns reflecting dihedral symmetry.

Applications and research: D-loops serve as objects of study in algebraic combinatorics and finite geometry, providing test cases for questions about associativity, inverse properties, and isotopy. They also appear in educational settings to illustrate nonassociative structures and have occasional connections to coding theory and cryptography through nontraditional algebraic frameworks.

See also: loop, quasigroup, dihedral group, isotopy, Moufang loop.