The Equivalence of the Identities: p(q.pr) = (pq.r) and pq.rp = p(qr.p)

Authors

  • Garba G. Zaku Department of Mathematics, University of Jos, Jos, Nigeria. Author
  • Lois A. Ademola Department of Mathematics, University of Jos, Jos, Nigeria. Author

DOI:

https://doi.org/10.62054/ijdm/0203.06

Abstract

Moufang loop <L, .>  is defined as a loop that satisfies any one of the identities: pq.rp =(p.qr)p, pq.rp= p(qr.p), (pq.r)q=p(q.rq) or p(q.pr)=(pq.p)r. This definition assumes the equivalence of these identities. We had earlier provided the proof of the equivalence of two of these identities: pq.rp =(p.qr)p and (pq.r)q=p(q.rq)  in [5], using a simple algebraic method and manipulation. In this paper we continue with our style of proof by providing the proof of the equivalence of another two of these identities: p(q.pr)=(pq.p)r and p(qr.p)=pq.rp. 

Author Biography

  • Lois A. Ademola, Department of Mathematics, University of Jos, Jos, Nigeria.

    Department of Mathematics,

    University of Jos, Jos, Nigeria.

References

Bruck R. H. (1971). A Survey of Binary Systems, Springer-Verlag, New York.

Drapal A. (2010). A Simplified Proof of Moufang’s Theorem, Proceedings of the American Mathematical Society, 139(1), p93-98.

Moufang R. (1935). Zur Struktur von Alternativkörpern, Math. Ann. 110, 416-430.

Pflugfelder H. O. (1990). Quasigroups and Loops: Introduction, Sigma Series in Pure Mathematics 7, Heldermann Verlag Berlin.

Zaku, G. G. and Jelten, N. B. (2023). The Equivalence of the Identities: amd . International Journal of Innovative Mathematics, Statistics & Energy Policies, 11(1) 36-42.

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Published

2025-09-28

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Issue

Vol. 2 No. 3 (2025): International Journal of Development Mathematics (IJDM)

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How to Cite

The Equivalence of the Identities: p(q.pr) = (pq.r) and pq.rp = p(qr.p). (2025). International Journal of Development Mathematics (IJDM), 2(3), 094-102. https://doi.org/10.62054/ijdm/0203.06