On the Coset laws for skew lattices in rings

Karin Cvetko-Vah, João Pita Costa

Skew lattices are the most successful generalization of lattices to the noncommutative case to date. Roughly speaking, each skew lattice can be seen as a lattice of rectangular bands. A coset decomposition can be given to each pair of comparable maximal rectangular bands. The internal structure of skew lattices is revealed by their coset structure. In the present paper we study the coset structure of skew lattices in rings and present certain coset laws that describe the connections among the coset decompositions given by distinct pairs of maximal rectangular bands.