Discussiones Mathematicae General Algebra and Applications 22(2) (2002) 199-216
doi: 10.7151/dmgaa.1058

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FREE ABELIAN EXTENSIONS IN THE CONGRUENCE-PERMUTABLE VARIETIES

Pavel Zhdanovich

Volgograd State Pedagogical University, Russia
e-mail:
comlab@vspu.ru

Abstract

We obtain the construction of free abelian extensions in a congurence-permutable variety V using the construction of a free abelian extension in a variety of algebras with one ternary Mal'cevoperation and a monoid of unary operations. We also use this construction to obtain a free solvable V-algebra.

Keywords: abelian extension, solvable algebra, congurence-permutable variety.

2000 Mathematics Subject Classification: 08B10, 08B20.

References

[1]
V.A. Artamonov, Magnus representation in congruence modular varieties (Russian), Sibir. Mat. Zh. 38 (1997), 978-995 (English transl. in Siberian Math. J. 38 (1997), 842-859.).
[2]
V.A. Artamonov and S. Chakrabarti, Free solvable algebra in a general congruence modular variety, Comm. Algebra 24 (1996), 1723-1735.
[3]
S. Chakrabarti, Homomorphisms of free solvable algebras with one ternary Mal'cev operation (Russian), Uspehi Mat. Nauk 48 (1993), no. 3, 207-208.
[4]
R. Freese and R. McKenzie, Commutator theory for congruence modular varieties, Cambridge Univ. Press, Cambridge 1987.
[5]
G. Grätzer, Universal Algebra (2nd ed.) Springer-Verlag, New York 1979.
[6]
A.G. Pinus, Congruence Modular Varieties (Russian), Irkutsk State University, Irkutsk 1986.
[7]
A.P. Zamyatin, Varieties with Restrictions on the Congruence Lattice (Russian), Ural State University, Sverdlovsk 1987.
[8]
P.B. Zhdanovich, Free Abelian extensions of ⟨p, S⟩-algebras, p. 73-80 in the book "Universal Algebra and its Applications", Proceedings of the Skornyakov Conference (Volgograd Ped. Univ., 1999), Izdat. "Peremena", Volgograd 2000.
[9]
P.B. Zhdanovich, Free Abelian extensions of Sp-permutable algebras (Russian), Chebyshevski Sbornik 3 (2002), No. 1 (3), 49-71.

Received 23 December 2002
Revised 13 January 2003
Revised 13 February 2003