Optimal Mal’tsev conditions for congruence modular varieties, Algebra Universalis 53

Abstract. For varieties, congruence modularity is equivalent to the tolerance intersection property, TIP in short. Based on TIP, it was proved in [5] that for an arbitrary lattice identity implying modularity (or at least congruence modularity) there exists a Mal'tsev condition such that the identity holds in congruence lattices of algebras of a variety if and only if the variety satises the corresponding Mal'tsev condition. However, the Mal'tsev condition con-structed in [5] is not the simplest known one in general. Now we improve this result by constructing the best Mal'tsev condition and various related condi-tions. As an application, we give a particularly easy new proof of Freese and Jonsson [11] stating that modular congruence varieties are Arguesian, and we strengthen this result by replacing "Arguesian " by "higher Arguesian " in the sense of Haiman [18]. We show that lattice terms for congruences of an arbi-trary congruence modular variety can be computed in two steps: the rst step mimics the use of congruence distributivity while the second step corresponds to congruence permutability. Particular cases of this result were known; the present approach using TIP is even simpler than the proofs of the previous partial results. 1