Higher homotopies between quasigroups (Q1734231)

The present paper is primarily intended as a study of the dual concept of higher homotopies. We remember that the algebraic concept of an isotopy, as a triple of bijections that generalizes the notion of an isomorphism, was introduced by \textit{A. A. Albert} for linear algebras [Ann. Math. (2) 43, 685--707 (1942; Zbl 0061.04807)]. He applied it to quasigroups in the article [\textit{A. A. Albert}, Trans. Am. Math. Soc. 54, 507--519 (1943; Zbl 0063.00039)]. In this article, he quotes the paper by \textit{N. Jacobson} [Duke Math. J. 3, 544--548 (1937; Zbl 0018.05005)] and says the following: ``A theory of non-associative algebras has been developed without any assumption of a substitute for the associative law, and the basic structure properties of such algebras have been shown to depend upon the possession of almost these same properties by related associative algebras. It seems natural then to attempt to obtain an analogous treatment of quasigroups. We shall present the results here''. This article contains the following sections. Section 1, Introduction. In Section 2, the authors study: Homotopies. In it, they present definitions, lemmas and Theorem 2.4. In Section 3, they study: Higher homotopies. In Section 4, the authors present: Principal higher isotopies. In Section 5, the authors study: Homotopy/higher homotopy duality. In Section 6, the authors present: Unit elements in domains. Here, they consider higher homotopies whose domain is a loop with identity element, or a quasigroup with a left or right unit element. In Section 7, they study: Counting higher homotopies. In this section, the main result is Theorem 7.4. In Section 8, they present: The category of higher homotopies. In Section 9, the authors study Moufang elements. In this section, the central result is Theorem 9.4.