The Equivalence of the Identities: p(q.pr) = (pq.r) and pq.rp = p(qr.p)
DOI:
https://doi.org/10.62054/ijdm/0203.06Abstract
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.
Riferimenti bibliografici
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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