Characterization of Groups, Loops, and Closure Conditions

Abstract

Groupoids, quasigroups, loops, semigroups, and groups are considered in this chapter. Closure conditions, isotopy, and groups are characterized by various identities. Functional equations arising out of Bol, Moufang, and extra loops are considered. Mediality, the left inverse property, Steiner loops, and generalized bisymmetry are treated. This chapter is devoted to algebraic identities (and their generalizations leading to the involvement of functional equations) connected to groups and well-known special loops such as Bol, Moufang, left inverse property (l.i.p.), and weak inverse property (w.i.p.). An identity in a binary system (such as transitivity, bisymmetry, distributivity, Bol, Moufang, etc.) induces a generalized identity—a functional equation—in a class of quasigroups. We treat several of them in this chapter. First we give some notation and definitions.KeywordsAbelian GroupFunctional EquationClosure ConditionMoufang LoopInverse PropertyThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.