Skew Lattice - The Coset Structure of Skew Lattices

The Coset Structure of Skew Lattices

A nonrectangular skew lattice is covered by its maximal primitive skew lattices: given comparable -classes in, forms a maximal primitive subalgebra of and every -class in lies in such a subalgebra. The coset structures on these primitive subalgebras combine to determine the outcomes and at least when and are comparable under . It turns out that and are determined in general by cosets and their bijections, although in a slightly less direct manner than the -comparable case. In particular, given two incomparable D-classes A and B with join D-class J and meet D-class in, interesting connections arise between the two coset decompositions of J (or M) with respect to A and B. (See Section 3.)

Thus a skew lattice may be viewed as a coset atlas of rectangular skew lattices placed on the vertices of a lattice and coset bijections between them, the latter seen as partial isomorphisms between the rectangular algebras with each coset bijection determining a corresponding pair of cosets. This perspective gives, in essence, the Hasse diagram of the skew lattice, which is easily drawn in cases of relatively small order. (See the diagrams in Section 3 above.) Given a chain of D-classes in, one has three sets of coset bijections: from A to B, from B to C and from A to C. In general, given coset bijections and, the composition of partial bijections could be empty. If it is not, then a unique coset bijection exists such that . (Again, is a bijection between a pair of cosets in and .) This inclusion can be strict. It is always an equality (given ) on a given skew lattice S precisely when S is categorical. In this case, by including the identity maps on each rectangular D-class and adjoining empty bijections between properly comparable D-classes, one has a category of rectangular algebras and coset bijections between them. The simple examples in Section 3 are categorical.