Not Every Osborn Loop Is Universal

arXiv:1003.1378v1 [math.GM] 6 Mar 2010 Not Every Osborn Loop Is Universal∗† T. G. Jaiy´eo.l´a‡ Department of Mathematics, Obafemi Awolowo University, Ile Ife 220005, Nigeria. jaiyeolatemitope@yahoo.com tjayeola@oauife.edu.ng J. O. Ad´en´ıran Department of Mathematics, University of Agriculture, Abeokuta 110101, Nigeria. ekenedilichineke@yahoo.com adeniranoj@unaab.edu.ng Abstract Michael Kinyon’s 2005 open problem, based on the universality of Osborn loops is solved. It is shown that not every Osborn loop is universal. 1 Introduction The universality of Osborn loops was raised as an open problem by Michael Kinyon [3] in 2005. Up to this present moment, the problem is still open and it is our aim in this study to lay it to rest in this note. It will be shown here that not every Osborn loop is universal. A loop is called an Osborn loop if it obeys the identity x(yz · x) = x(yxλ · x) · zx. An Osborn loop is said to be universal if all its loop isotopes are Osborn loops. The most popularly known varieties of Osborn loops are CC-loops, Moufang loops, VD-loops and universal weak inverse property loops. All these four varieties of Osborn loops are universal and this is what makes the problem interesting. For a comprehensive and detailed introduction and literature review of the subject of Osborn loops and universality, readers should check Jaiy´eo.l´a and Ad´en´ıran [2]. The notations and symbols used here are exactly those adopted in [2]. ∗2000 Mathematics Subject Classification. Primary 20NO5 ; Secondary 08A05 †Keywords and Phrases : Osborn loops, universality of Osborn loops ‡All correspondence to be addressed to this author. 1 2 Main Result Theorem 2.1 Not every Osborn loop is universal. Proof Huthnance [1] gave an example of an Osborn loop as follows. Let H = Z × Z × Z. Define a binary operation ⋆on H by : [2i, k, m] ⋆[2j, p, q] = [2i + 2j, k + p −ij(2j −1), q + m −ij(2j −1)] [2i + 1, k, m] ⋆[2j, p, q] = [2i + 2j + 1, k + p −ij(2j −1) −j2 + j, q + m −ij(2j −1) −j2] [2i, k, m] ⋆[2j + 1, p, q] = [2i + 2j + 1, m + p −ij(2j + 1), q + k −ij(2j + 1)] [2i + 1, k, m] ⋆[2j + 1, p, q] = [2i + 2j + 2, m + p −ij(2j + 1) −j2 + j, q + k −ij(2j + 1) −j2] for all i, j, k, m, p, q ∈Z. Assuming that (H, ⋆) is a universal Osborn loop, then it should obey the identity v · vv = vλ\v · v in Lemma 3.12 of [2]. Let v = [2i + 1, k, m]. Then, by direct computation, we have v · vv = [6i + 3, m + 2k −10i3 −12i2 −2i, 2m + k −10i3 −12i2 −i −1] and vλ\v · v = [6i + 3, m + 2k −14i3 −18i2 −7i −1, 2m + k −14i3 −16i2 −6i −1]. So, v · vv ̸= vλ\v · v. Thus, (H, ⋆) is not a universal Osborn loop. 3 Concluding Remarks and Future Studies Theorem 2.1 completely lays to rest the open problem of whether or not Osborn loops are generally universal. Kinyon [3] went furthermore to ask the following question, if in case not every Osborn loop is universal. • Does there exist a proper Osborn loop with trivial nucleus? The answer to this question is due for future study. References [1] Huthnance Jr., E.D., A theory of generalised Moufang loops, Ph.D. thesis, Georgia Institute of Technology, 1968. [2] Jaiy´eo.l´a, T.G., and Ad´en´ıran, J.O., New identities in universal Osborn loops, Quasi- groups And Related Systems, 17 (2009), to appear. [3] Kinyon, M.K., A survey of Osborn loops, Milehigh conference on loops, quasigroups and non-associative systems, University of Denver, Denver, Colorado, 2005. 2

This content is AI-processed based on open access ArXiv data.