It is shown that a variety generated by a nilpotent A-loop has a decidable equational (quasiequational ) theory. Thereby the question posed by A. I. Mal’tsev in [6] is answered in the negative, and moreover, a finitely presented nilpotent A-loop has a decidable word problem.
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References
R. C. Lyndon, “Two notes on nilpotent groups,” Proc. Am. Math. Soc., 3, 579–583 (1952).
H. Neumann, Varieties of Groups, Springer, Berlin (1967).
S. Oates and M. B. Powell, “Identical relations in finite groups,” J. Alg., 1, 11–39 (1964).
T. Evans, “Identities and relations in commutative Moufang loops,” J. Alg., 31, 508–513 (1974).
V. Ursu, “On identities of nilpotent Moufang loops,” Rev. Roum. Math. Pures Appl., 45, No. 3, 537–548 (2000).
A. Mal’tsev, “Identical relations on varieties of quasigroups,” Mat. Sb., 69(111), No. 1, 3–12 (1966).
V. D. Belousov, Foundations of the Theory of Quasi-Groups and Loops [in Russian], Nauka, Moscow (1967).
R. H. Bruck, A Survey of Binary Systems, Springer, Berlin (1958).
R. H. Bruck and L. J. Paige, “Loops whose inner mappings are automorphisms,” Ann. Math. (2), 63, 308–323 (1956).
Quasigroups and Loops: Theory and Applications, Sigma Ser. Pure Math., 8, Heldermann, Berlin (1990).
G. C. McKinsey, “The decision problem for some classes of sentences without quantifiers,” J. Symb. Log., 8, 61–76 (1943).
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In memory of V. A. Gorbunov
Translated from Algebra i Logika, Vol. 49, No. 4, pp. 479–497, July–August, 2010.
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Kowalski, A.V., Ursu, V.I. An equational theory for a nilpotent A-loop. Algebra Logic 49, 326–339 (2010). https://doi.org/10.1007/s10469-010-9099-0
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DOI: https://doi.org/10.1007/s10469-010-9099-0