Minimal formations of universal algebras

Wenbin Guo; K.P. Shum

Discussiones Mathematicae - General Algebra and Applications (2001)

  • Volume: 21, Issue: 2, page 201-205
  • ISSN: 1509-9415

Abstract

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A class ℱ of universal algebras is called a formation if the following conditions are satisfied: 1) Any homomorphic image of A ∈ ℱ is in ℱ; 2) If α₁, α₂ are congruences on A and A / α i ∈ ℱ , i = 1,2, then A/(α₁∩α₂) ∈ 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.

How to cite

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Wenbin Guo, and K.P. Shum. "Minimal formations of universal algebras." Discussiones Mathematicae - General Algebra and Applications 21.2 (2001): 201-205. <http://eudml.org/doc/287759>.

@article{WenbinGuo2001,
abstract = {A class ℱ of universal algebras is called a formation if the following conditions are satisfied: 1) Any homomorphic image of A ∈ ℱ is in ℱ; 2) If α₁, α₂ are congruences on A and $A/α_\{i\} ∈ ℱ$, i = 1,2, then A/(α₁∩α₂) ∈ 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.},
author = {Wenbin Guo, K.P. Shum},
journal = {Discussiones Mathematicae - General Algebra and Applications},
keywords = {universal algebra; congruence; formation; minimal subformation},
language = {eng},
number = {2},
pages = {201-205},
title = {Minimal formations of universal algebras},
url = {http://eudml.org/doc/287759},
volume = {21},
year = {2001},
}

TY - JOUR
AU - Wenbin Guo
AU - K.P. Shum
TI - Minimal formations of universal algebras
JO - Discussiones Mathematicae - General Algebra and Applications
PY - 2001
VL - 21
IS - 2
SP - 201
EP - 205
AB - A class ℱ of universal algebras is called a formation if the following conditions are satisfied: 1) Any homomorphic image of A ∈ ℱ is in ℱ; 2) If α₁, α₂ are congruences on A and $A/α_{i} ∈ ℱ$, i = 1,2, then A/(α₁∩α₂) ∈ 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.
LA - eng
KW - universal algebra; congruence; formation; minimal subformation
UR - http://eudml.org/doc/287759
ER -

References

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  1. [1] L.A. Artamonov, V.N. Sali, L.A. Skorniakov, L.N. Shevrin and E.G. Shulgeifer, General Algebra (Russian), vol. II, Izd. 'Nauka', Moscow 1991. 
  2. [2] D.W. Barnes, Saturated formations of soluable Lie algebras in characteristic zero, Arch. Math., 30 (1978), 477-480. Zbl0365.17007
  3. [3] D.W. Barnes and H.M. Gastineau-Hills, On the theory of soluble Lie algebras, Math. Z., 106 (1969), 343-353. Zbl0164.03701
  4. [4] K. Doerk and T.O. Hawkes, Finite soluable groups, Walter de Gruyter & Co., Berlin 1992. Zbl0753.20001
  5. [5] A.I. Mal˘cev, Algebraic systems (Russian), Izd. 'Nauka', Moscow 1970. 
  6. [6] L.A. Shemetkov, Formations of finite groups (Russian), Izd. 'Nauka', Moscow 1978. 
  7. [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). Zbl0573.08007
  8. [8] L.A. Shemetkov and A.N. Skiba, Formations of algebraic systems (Russian), Izd. 'Nauka', Moscow 1989. Zbl0667.08001

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