Discussiones Mathematicae General Algebra and Applications 21(2) (2001)201-205
doi: 10.7151/dmgaa.1037

[BIBTex] [PDF] [PS]

MINIMAL FORMATIONS OF UNIVERSAL ALGEBRAS

Wenbin Guo

Department of Mathematics,
Xuzhou Normal University, Xuzhou 221009, P. R. China
e-mail:
yzgwb@pub.yz.jsinfo.net

K.P. Shum

Department of Mathematics,
The Chinese University of Hong Kong, Shatin, N.T.
Hong Kong, China (SAR)
e-mail:
kpshum@math.cuhk.edu.hk

Abstract

A class F of universal algebras is called a formation if the following conditions are satisfied: 1) Any homomorphic image of A Î F is in F; 2) If a1, a2 are congruences on A and A/ai Î F, i = 1,2, then A/(a1Ça2) Î F. We prove that any formation generated by a simple algebra with permutable congruences is minimal, and hence any formation containing a simple algebra, with permutable congruences, contains a minimum subformation. This result gives a partial answer to an open problem of Shemetkov and Skiba on formations of finite universal algebras proposed in 1989.

Keywords: universal algebra; congruence; formation; minimal subformation.

2000 AMS Mathematics Subject Classifications: 03C05, 08B05, 08B10.

References

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[2] D.W. Barnes, Saturated formations of soluable Lie algebras in characteristic zero, Arch. Math., 30 (1978), 477-480.
[3] D.W. Barnes and H.M. Gastineau-Hills, On the theory of soluble Lie algebras, Math. Z., 106 (1969), 343-353.
[4] K. Doerk and T.O. Hawkes, Finite soluable groups, Walter de Gruyter & Co., Berlin 1992.
[5] A.I. Mal˘cev, Algebraic systems (Russian), Izd. "Nauka", Moscow 1970.
[6] L.A. Shemetkov, Formations of finite groups (Russian), Izd. "Nauka", Moscow 1978.
[7] L.A. Shemetkov, The product of any formation of algebraic systems (Russian), Algebra i Logika, 23 (1984), 721-729. (English transl.: Algebra and Logic 23 (1985), 489-490)
[8] L.A. Shemetkov and A.N. Skiba, Formations of algebraic systems (Russian), Izd. "Nauka", Moscow 1989.

Received 29 March 2001
Revised 25 September 2001