On the relationship between the rotation construction and ordered Abelian groups

By analogy with the usual extension of the group operation from the positive cone of an ordered Abelian group into the whole group, a construction—called symmetrization—is defined and it is related to the rotation construction [Jenei, On the structure of rotation-invariant semigroups, Archive for Mathematical Logic 42 (2003) 489–514]. Symmetrization turns out to be a kind of dualized rotation. A characterization is given for the left-continuous t-conorms for which their symmetrization is a uninorm. As a by-product a new family of involutive uninorms is introduced.