Flat coset decompositions of skew lattices

Skew lattices are non-commutative generalizations of lattices, and the cosets are the building blocks of skew lattices. Every skew lattice embeds into a direct product of a left-handed skew lattice and a right-handed skew lattice. It is therefore natural to consider the flat coset decompositions, i.e. decompositions of a skew lattice into right and left cosets. In the present paper we discuss such decompositions, their properties and the relation to the coset laws for cancellative and symmetric skew lattices.