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.
@article{bwmeta1.element.bwnjournal-article-doi-10_7151_dmgaa_1052,
author = {Ivan Chajda and Radom\'\i r Hala\v s},
title = {Congruence submodularity},
journal = {Discussiones Mathematicae - General Algebra and Applications},
volume = {22},
year = {2002},
pages = {131-139},
zbl = {1037.08002},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_7151_dmgaa_1052}
}
Ivan Chajda; Radomír Halaš. Congruence submodularity. Discussiones Mathematicae - General Algebra and Applications, Tome 22 (2002) pp. 131-139. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_7151_dmgaa_1052/
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