Residuated skew lattices

Skew lattices are one of the most successful non-commutative generalizations of lattices. Motivated by the study of residuation on ordered structures and that of skew Boolean algebras and skew Heyting algebras, in this paper, we introduce the concept of residuated skew lattices as a new non-commutative version of residuated lattices. The axiomatization and localization for residuated skew lattices are obtained. Moreover, a characterization of residuation on skew lattices via adjointness is presented. Meanwhile, three special subvarieties: distributive residuated skew lattices, skew MTL-algebras and skew BL-algebras, are investigated. In particular, we show that every skew Heyting algebra is a reduct of a special residuated skew lattice.