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All congruence lattice identities implying modularity have Mal’tsev conditions. (English) Zbl 1091.08007
For an arbitrary lattice identity implying congruence modularity, a Mal’tsev condition is given such that the identity holds in congruence lattices of algebras of a variety if and only if the variety satisfies the corresponding Mal’tsev condition. This Mal’tsev condition is formulated by means of Day’s condition for congruence modularity and the original identity $$p\leq q$$ (stronger than modularity) such that joins in $$p$$ and $$q$$ are substituted by a certain number of relational products and a Mal’tsev condition for the resulting identity is derived by the standard algorithm of R. Wille.

##### MSC:
 08B10 Congruence modularity, congruence distributivity 08B05 Equational logic, Mal’tsev conditions
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