The Theory of Linear Lattices

We make a study of lattices representable by commuting equivalence relations, which we call linear lattices. We develop a proof theory for implications valid in linear lattices, which differs from classical proof theories in that its deductions transform representative graphs rather than well-formed formulas. We then use graph-theoretic arguments to establish a duality theorem and a normal form theorem for this proof theory.

We introduce a matroid analogue of the graph proof theory, and find the class of lattices to which it applies. This turns out to be the class of lattices representable by what we call pseudo-congruences on a Mal'cev algebra.

For completeness, we review some known results about lattices of ordinary congruences on a Mal'cev algebra, showing how they may be reformulated in the present terminology.