congruence permutable varieties
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2010 ◽  
Vol 89 (1) ◽  
pp. 105-126 ◽  
Author(s):  
MARTYN QUICK ◽  
N. RUŠKUC

AbstractFor an algebraic structure A denote byd(A) the smallest size of a generating set for A, and letd(A)=(d(A),d(A2),d(A3),…), where An denotes a direct power of A. In this paper we investigate the asymptotic behaviour of the sequenced(A) when A is one of the classical structures—a group, ring, module, algebra or Lie algebra. We show that if A is finite thend(A) grows either linearly or logarithmically. In the infinite case constant growth becomes another possibility; in particular, if A is an infinite simple structure belonging to one of the above classes thend(A) is eventually constant. Where appropriate we frame our exposition within the general theory of congruence permutable varieties.


2008 ◽  
Vol 85 (1) ◽  
pp. 59-74 ◽  
Author(s):  
BRIAN A. DAVEY ◽  
MARCEL JACKSON ◽  
MIKLÓS MARÓTI ◽  
RALPH N. MCKENZIE

AbstractWe give a new proof that a finitely generated congruence-distributive variety has finitely determined syntactic congruences (or, equivalently, term finite principal congruences), and show that the same does not hold for finitely generated congruence-permutable varieties, even under the additional assumption that the variety is residually very finite.


Author(s):  
B. J. Gardner

AbstractA variant of Kurosh-Amitsur radical theory is developed for algebras with a collection of (finitary) operations ω, all of which are idempotent, that is satisfy the condition ω(x, x,…,x) =x. In such algebras, all classes of any congruence are subalgebras. In place of a largest normal radical subobject, a largest congruence with radical congruence classes is considered. In congruence-permutable varieties the parallels with conventional radical theory are most striking.


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