  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.
