The modularity condition for semi-t-operators

The aim of this paper is mainly to solve the functional equations given by the modularity condition. Several years ago, the modularity equations were discussed in families of certain commutative operations, such as t-norms, t-conorms, uninorms, 2-uninorms and t-operators. In this study, we continue the investigation of this same topic by focusing on semi-t-operators, which are generalization of t-operators by omitting commutativity. Our results are not only the full characterization of this modularity condition for semi-t-operators with continuous underlying operators, but also generalize the case for t-operators under certain conditions.