# Distributivity in skew lattices

Research paper by Michael Kinyon, Jonathan Leech, João Pita Costa

Indexed on: 05 May '15Published on: 05 May '15Published in: Semigroup Forum

#### Abstract

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 by the behavior of the natural partial order $$\ge$$ on elements in chains of comparable $${\mathcal {D}}$$-classes, $$A>B>C$$, with particular attention given to midpoints $$b$$ of chains $$a > b > c$$ where $$a \in A$$, $$b \in B$$ and $$c \in C$$. Since distributive skew lattices are linearly distributive and have distributive maximal lattice images (but not conversely in general), we give criteria that guarantee that skew lattices with both properties are distributive. In particular symmetric skew lattices (where $$x\wedge y = y\wedge x$$ if and only if $$x\vee y = y\vee x$$) that have both properties are distributive.