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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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