On left and right uninorms on a finite chain

The main concern of this paper is to introduce and characterize the class of operators on a finite chain L, having the same properties of pseudosmooth uninorms but without commutativity. Moreover, in this case it will only be required the existence of a one-side neutral element. These operators are characterized as combinations of AND and OR operators of directed algebras (smooth t-norms and smooth t-conorms) and the case of pseudosmooth uninorms is retrieved for the commutative case.