On the Coset Structure of Skew Lattices

Skew lattices are the mostly studied noncommutative generalization of lattices. The class of skew lattices can be seen as an algebraic category. It models an algebraic theory in the category of Sets where the Green's relation D is a congruence describing an adjunction to the category of Lattices. In this talk we will discuss the relevance of this approach, revisit some known decompositions and relate the order structure of a skew lattice with its coset structure that describes the internal coset decomposition of the respective skew lattice. Moreover, we will present the ideals in skew lattices and the intersections of this theory with the coset structure of a skew lattice.