I am surprised that nobody came up with median algebras. That is, algebras that are equipped with a single ternary (fundamental) operation (see http://en.wikipedia.org/wiki/Median_algebra for a definition and some references). They generalize distributive lattices since the median function $(x \vee y) \wedge (y \vee z) \wedge (z \vee x)$ of any distributive lattice gives rise to a median algebra. However, the concept is more subtle and the category of median algebras is not equivalent to that of distributive lattices.

Median algebras are also interesting since they have beautiful duality with so-called Isbell spaces (first described by, you guessed correclty, John Isbell). That is, bounded Priestley spaces that are also equipped with certain (unary) complement operation.