Distributivity in skew lattices

Distributive skew lattices satisfying $x\wedge (y\vee z)\wedge x = (x\wedge y\wedge x) \vee (x\wedge z\wedge x)$ and its dual are studied, along with the larger class of linearly distributive skew lattices, whose totally preordered subalgebras are distributive. Linear distributivity is characterized in terms of the behavior of the natural partial order between comparable $\DD$-classes. This leads to a second characterization in terms of strictly categorical skew lattices. Criteria are given for both types of skew lattices to be distributive.