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Loops and quasigroups: Aspects of current work and prospects for the future. (English) Zbl 1038.20055
This paper presents the contents of the opening address by the author at the international conference Loops ’99 held in Prague from July 27 till August 1, 1999. Recent trends and future prospects of the theory of loops and quasigroups are discussed, based upon the author’s broad and deep vision of the field. Main attention is paid to those areas which have connections to other parts of mathematics and to applications outside mathematics. The main topics presented are nets and homotopy (including Lindenbaum-Tarski duality), transversals (with connections to coding theory), octonions, topology, and knots (with quandles, i.e. right distributive right quasigroups, as invariants determining knots, and with Conway algebras), and representation theory (with permutation representations, modules, and characters).

MSC:
20N05 Loops, quasigroups
22A30 Other topological algebraic systems and their representations
53A60 Differential geometry of webs
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