On Skew Lattices

Skew lattices are the most studied non-commutative analogue of lattices and may be viewed as double bands of idempotents in a semigroup context. The class of skew lattices forms an algebraic category with several interesting properties. They model an algebraic theory in the category of sets where the Green's congruence D is used to define an adjuction to the category of Lattices. In this talk we are going to present the relations between the varieties of strongly symmetric skew lattices, cancellative skew lattices and symmetric skew lattices. We generalize the parallelogram laws stated earlier for cancellative skew lattices. These will characterize each one of the varieties. Later we will present some combinatoric results that follow from the parallelogram laws and resemble the Lagrange Theorem for groups.