Skew residuated lattices

We replace the so-called adjointness in the definition of residuated lattice by its strict version where inequalities are replaced by equalities. We prove that such structures, called skew residuated lattices, can be characterized as lattices with certain involutions in principal filters. Since skew residuated lattices have the cancellation property, they are close to divisibility loops introduced by B. Bosbach in 1988. We show under what condition can skew residuated lattices be represented by such loops.