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Given a variety V with a constant 0 in its type and a lattice identity p ≤ q, we say that p ≤ q holds for congruences in V at 0 if the p-block of 0 is included in the q-block of 0 for all substitutions of congruences of V-algebras for the variables of p and q. Varieties that are congruence modular at 0 are characterized by a Mal'tsev condition. This result generalizes the classical characterization of congruence modularity by Day terms.

We prove the following theorem. Let (a 1 , . . . , am, c 12 , . . . , c 1m ) be a spanning von Neumann m-frame of a modular lattice L, and let (u 1 , . . . , un, v 12 , . . . , v 1n ) be a spanning von Neumann n-frame of the interval [0, a 1 ]. Assume that either m ≥ 4, or L is Arguesian and m ≥ 3. Let R * denote the coordinate ring of (a 1 , . . . , am, c 12 , . . . , c 1m ). If n ≥ 2, then there is a ring S * such that R * is isomorphic to the ring of all n × n matrices over S * . If n ≥ 4 or L is Arguesian and n ≥ 3, then we can choose S * as the coordinate ring of (u 1 , . . . , un, v 12 , . . . , v 1n ).

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We define the order-congruence distributivity at 0 and ordercongruence n-distributivity at 0 of ordered algebras with a nullary operation 0. These notions are generalizations of congruence distributivity and congruence n-distributivity. We prove that a class of ordered algebras with a nullary operation 0 closed under taking subalgebras and direct products is order-congruence distributive at 0 iff it is ordercongruence n-distributive at 0. We also characterize such classes by a Mal'tsev condition.