Abstract
This chapter discusses the analysis of splitting lattices and congruence modularity. It has been shown that the modular law is a consequence of weaker lattice theoretical assumptions on the congruence variety of an arbitrary variety of algebras. These results generated a conjecture, viz., if the congruence variety of a variety of algebras satisfies any nontrivial lattice identity, then it is already congruence modular. A main characteristic of splitting lattices is that each comes paired with a (conjugate) equation so that every variety of lattices either satisfies this equation or contains the paired splitting lattice but not both. This allows one to alternate between semantical and syntactical arguments as best befits the situation at hand. The chapter analyzes a class, l1 of splitting lattices such that for every S in l1, if the congruence variety (of a variety of algebras) satisfies the conjugate equation of S, then the variety is already congruence modular.
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