The Shifting Lemma and shifting lattice identities

Abstract.

Gumm [6] used the Shifting Lemma with high success in congruence modular varieties. Later, some analogous diagrammatic statements, including the Triangular Scheme from [1] were also investigated. The present paper deals with the purely lattice theoretic underlying reason for the validity of these lemmas. The shift of a lattice identity, a special Horn sentence, is introduced. To any lattice identity λ and to any variable y occurring in λ we introduce a Horn sentence S(λ, y). When S(λ, y) happens to be equivalent to λ, we call it a shift of λ. When λ has a shift then it gives rise to diagrammatic statements resembling the Shifting Lemma and the Triangular Scheme. Some known lattice identities will be shown to have a shift while some others have no shift.