Abstract
An equivalence \(\sim \) upon a loop is said to be multiplicative if it satisfies \(x\sim y\), \(u \sim v \Rightarrow xu \sim yv\). Let X be a set with elements \(x\ne y\) and let \(\sim \) be the least multiplicative equivalence upon a free loop F(X) for which \(x\sim y\). If \(a,b\in F(X)\) are such that \(a\ne b\) and \(a\sim b\), then neither \(a\backslash c \sim b\backslash c\) nor \(c/a \sim c/b\) is true, for every \(c\in F(X)\).
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References
Solution II to problem 10888, Am. Math. Mon. 110(4), 347 (2003)
Bates, G.E., Kiokemeister, F.: A note on homomorphic mappings of quasigroups into multiplicative systems. Bull. Am. Math. Soc. 54, 1180–1185 (1948)
Evans, T.: On multiplicative systems defined by generators and relations. I. Normal form theorems. Proc. Camb. Philos. Soc. 47, 637–649 (1951)
Glauberman, G.: On loops of odd order. J. Algebra 1, 374–396 (1964)
Pflugfelder, H.O.: Quasigroups and Loops: Introduction. Heldermann, Berlin (1990)
Smith, J.D.H.: An Introduction to Quasigroups and Their Representations. Chapman & Hall/CRC, Boca Raton (2007)
Acknowledgements
I thank Michael K. Kinyon for informing me about constructions of multiplicative equivalences that do not yield a loop or quasigroup and are cited in this paper. I also thank the anonymous referee for several suggestions how to add perspective to the paper.
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Drápal, A. On multiplicative equivalences that are totally incompatible with division. Algebra Univers. 80, 32 (2019). https://doi.org/10.1007/s00012-019-0605-5
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DOI: https://doi.org/10.1007/s00012-019-0605-5