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DEMONSTRATIO MATHEMATICAVol. XXVIINo 3 - 41994Edgar G. Goodaire, D . A . RobinsonS E M I - D I R E C T P R O D U C T S A N D BOL LOOPDedicatedto ProfessorTadeuszTraczyk1. IntroductionIf Κ is a loop 1 satisfying some loop-theoretic property V (usually described by means of identities), it is often of interest to extend Κ to loopsG which also satisfy V. In such situations, the following question is relevant: Given loops H and Κ with Κ satisfying a property V, can one usethe Cartesian product set Η χ Κ as the carrier set for various "semi-direct"product loops G — Η χ Κ which also satisfy property V and which have asubloop Κ isomorphic to Κ serving as the kernel of a (loop) homomorphismof G onto ΗΊIn §2 we discuss how and precisely under what circumstances (external)semi-direct products of loops can be formed, and show exactly when a loopis an (internal) semi-direct product of a pair of subloops. In this section, weinclude some examples, and show that some constructions in the literature— seemingly ad hoc — are in reality semi-direct products.In §3 we select a specific loop-theoretic property V, namely, that ofsatisfying the
Demonstratio Mathematica – de Gruyter
Published: Jul 1, 1994
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