Abstract
The Loewy rank of a modular latticeL of finite height is defined as the leastn for which there exista0=0<at, < ... <ar=1 inL such that each interval I[ai, ai+1] is a complemented lattice. In this paper, a generalized notion of Loewy rank is applied to obtain new results in the commutator theory of locally finite congruence modular varieties. LetV be a finitely generated congruence modular variety. We prove that every algebra inV has a largest nilpotent congruence and a largest solvable congruence. Moreover, there exist first order formulas which define these special congruences in every algebra ofV.
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