Distributive Skew Lattices

A new paper is out contributing to the study of distributivity in the context of skew lattices. It follows the discussion of the paper Cancellative Skew Lattices and reflects the work on distributivity in skew chains and its role in the study of a global distributivity. As expected there are many possible generalizations of this property in these noncommutative algebras and the reader can find the clarification on the relations between them. Furthermore, the later stage of this research shows the importance of the study of the coset structure of the skew lattice, referencing to the work presented in the paper On the coset structure of categorical skew lattices by Joao Pita Costa. Bellow follows the abstract and the references to this paper, submitted to the Semigroup Forum.

Distributivity in skew lattices

arXiv:1306.5598

Michael Kinyon, Jonathan Leech, Joao Pita Costa

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.