Geodesic
Some modifications of congruence permutability and dually congruence regular varietie
Discussiones Mathematicae. General Algebra and Applications, Tome 21 (2001) no. 2, pp. 165-174.

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It is well known that every congruence regular variety is n-permutable (in the sense of [9]) for some n ≥ 2. For the explicit proof see e.g. [2]. The connections between this n and Mal'cev type characterizations of congruence regularity were studied by G.D. Barbour and J.G. Raftery [1]. The concept of local congruence regularity was introduced in [3]. A common generalization of congruence regularity and local congruence regularity was given in [6] under the name "dual congruence regularity with respect to a unary term g". The natural problem arises what modification of n-permutability is satisfied by dually congruence regular varieties. The aim of this paper is to find out such a modification, to characterize varieties satisfying it by a Mal'cev type condition and to show connections with normally presented varieties (see e.g. [5], [8], [11]). The latter concept was introduced already by J. P≥onka under a different term; the names "normal identity" and "normal variety" were firstly used by E. Graczyńska in [8].
Keywords: congruence regularity, local congruence regularity, dual congruence regularity, local n-permutability 
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Chajda, Ivan; Eigenthaler, Günther. Some modifications of congruence permutability and dually congruence regular varietie. Discussiones Mathematicae. General Algebra and Applications, Tome 21 (2001) no. 2, pp. 165-174. http://geodesic.mathdoc.fr/item/DMGAA_2001_21_2_a2/

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