Structural Description of a Class of Involutive Uninorms via Skew Symmetrization

The main ‘philosophical’ outcome of this article is to demonstrate that the structural description of residuated lattices requires the use of the co-residuated setting. A construction, called skew symmetrization, which generalizes the well-known representation of an ordered Abelian group obtained from the positive (or negative) cone of the algebra is introduced here. Its definition requires leaving the accustomed residuated setting and entering the co-residuated setting. It is shown that every uninorm on [0,1] with an involution defined by the residual complement with respect to the unit and having the unit as the fixed point of the involution can be described as the skew symmetrization of its underlying t-norm or underlying t-conorm.