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Keywords:
derived operations; isotopy; quasigroups; position condition; regular universal identity; identities; $3$-basic quasigroups; universal identity; $3$-basic quasigroup identity
derived operations; isotopy; quasigroups; position condition; regular universal identity; identities; $3$-basic quasigroups; universal identity; $3$-basic quasigroup identity
Summary:
The paper deals with quasigroup identities under isotopies. The terminology is taken from [2], [3] and [4]. Stimulated by geometric illustrations, V. D. Belousov in [2] has presented two important identity properties and posed a question for which identities these properties are necessary and sufficient for the identity to be invariant under isotopies. Inspired by V. D. Belousov, G. Monoszová investigated in [6] one special kind of identities for which both Belousov's properties give necessary and sufficient conditions for the identity to be invariant under isotopies. Our purpose is to amend Belousov's properties to such ones which guarantee the identity invariance under isotopies for general identities. We also show a close connection between quasigroup identities invariant under isotopies and $3$-basic quasigroup identities.
References:
[1] V. D. Belousov: Foundations of the theory of quasigroups and loops. Moskva, 1967. (In Russian.) MR 0218483
[2] V. D. Belousov: Algebraic nets and quasigroups. Kishinev, 1971. (In Russian.)
[3] E. Brožíková: On 3-basic quasigroups and their congruences. Časopis pro pěst. matematiky 115, no. 1 (1990), 38-47. MR 1044012
[4] T. Evans: The word problem for abstract algebras. J. London Math. Soc. 26 (1951), 64-71. DOI 10.1112/jlms/s1-26.1.64 | MR 0038958 | Zbl 0042.03303
[5] T. Evans: On multiplicative systems defìned by generators and relations. Proc. Cambridge Philos. Soc. 47 (1951), 637-649. MR 0043764 | Zbl 0043.02001
[6] G. Monoszová: Some quasigroup identities invariant under quasigroup isotopies. Matem. issledovanija, Kishinev no. 102 (1988), 80-91. (In Russian.) MR 0947461
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