# zbMATH — the first resource for mathematics

An equational theory for a nilpotent $$A$$-loop. (English. Russian original) Zbl 1255.20059
Algebra Logic 49, No. 4, 326-339 (2010); translation from Algebra Logika 49, No. 4, 479-497 (2010).
Summary: It is shown that the variety generated by a nilpotent $$A$$-loop has a decidable equational (quasiequational) theory. Thereby the question posed by A. I. Mal’tsev [in Mat. Sb., N. Ser. 69(111), 3-12 (1966; Zbl 0202.31201)] is answered in the negative, and moreover, a finitely presented nilpotent $$A$$-loop has decidable word problem.

##### MSC:
 20N05 Loops, quasigroups 08B05 Equational logic, Mal’tsev conditions
Full Text:
##### References:
 [1] R. C. Lyndon, ”Two notes on nilpotent groups,” Proc. Am. Math. Soc., 3, 579–583 (1952). · Zbl 0047.25602 [2] H. Neumann, Varieties of Groups, Springer, Berlin (1967). · Zbl 0149.26704 [3] S. Oates and M. B. Powell, ”Identical relations in finite groups,” J. Alg., 1, 11–39 (1964). · Zbl 0121.27202 [4] T. Evans, ”Identities and relations in commutative Moufang loops,” J. Alg., 31, 508–513 (1974). · Zbl 0285.20058 [5] V. Ursu, ”On identities of nilpotent Moufang loops,” Rev. Roum. Math. Pures Appl., 45, No. 3, 537–548 (2000). · Zbl 0993.20043 [6] A. Mal’tsev, ”Identical relations on varieties of quasigroups,” Mat. Sb., 69(111), No. 1, 3–12 (1966). [7] V. D. Belousov, Foundations of the Theory of Quasi-Groups and Loops [in Russian], Nauka, Moscow (1967). · Zbl 0163.01801 [8] R. H. Bruck, A Survey of Binary Systems, Springer, Berlin (1958). · Zbl 0081.01704 [9] R. H. Bruck and L. J. Paige, ”Loops whose inner mappings are automorphisms,” Ann. Math. (2), 63, 308–323 (1956). · Zbl 0074.01701 [10] Quasigroups and Loops: Theory and Applications, Sigma Ser. Pure Math., 8, Heldermann, Berlin (1990). [11] G. C. McKinsey, ”The decision problem for some classes of sentences without quantifiers,” J. Symb. Log., 8, 61–76 (1943). · Zbl 0063.03864
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.