Geodesic
Congruence submodularity
Discussiones Mathematicae. General Algebra and Applications, Tome 22 (2002) no. 2, pp. 131-139

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We present a countable infinite chain of conditions which are essentially weaker then congruence modularity (with exception of first two). For varieties of algebras, the third of these conditions, the so called 4-submodularity, is equivalent to congruence modularity. This is not true for single algebras in general. These conditions are characterized by Maltsev type conditions.
Keywords: congruence lattice, modularity, congruence k-submodularity 
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Chajda, Ivan; Halaš, Radomír. Congruence submodularity. Discussiones Mathematicae. General Algebra and Applications, Tome 22 (2002) no. 2, pp. 131-139. http://geodesic.mathdoc.fr/item/DMGAA_2002_22_2_a2/