Weak distributive laws and their role in lattices of congruences and equational theories

By a result of Pigozzi and Kogalovskii, every algebraic latticeL having a completely join —irreducible top element can be represented as the lattice L(Σ) of equational theories extending some fixed theory Σ. Conversely, strengthening a recent result due to Lampe, we show that such a representationL=L(Σ) forcesL to satisfy the following condition: if the top element ofL is the join of a nonempty subsetB ofL then there are elementsb..., ε B such thata=(... (((b1 ∧a) ∨ b2) ∧a) ... ∨ bn) ∧a for alla ε L. In presence of modularity, this equation reduces to the identitya=(a ∧ b1) ∨ ... ∨ (a ∧ bn). Motivated by these facts, we study several weak forms of distributive laws in arbitrary lattices and related types of prime elements. The main tool for applications to universal algebra is a generalized version of Lampe's Zipper Lemma.