Geodesic
Minimal formations of universal algebras
Discussiones Mathematicae. General Algebra and Applications, Tome 21 (2001) no. 2, pp. 201-205

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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.
Keywords: universal algebra, congruence, formation, minimal subformation 
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Guo, Wenbin; Shum, K. Minimal formations of universal algebras. Discussiones Mathematicae. General Algebra and Applications, Tome 21 (2001) no. 2, pp. 201-205. http://geodesic.mathdoc.fr/item/DMGAA_2001_21_2_a4/