The modularity condition for semi-t-operators and semi-uninorms

The aim of this paper is mainly to solve the functional equations given by the modularity condition. Several years ago, the modularity equations for t-norms, t-conorms, uninorms and t-operators, which are commutative and associative, have been studied. Our investigations are motivated by modularity condition for generalizations of these operators by removing associativity or commutativity. In this work, the following main conclusions are proved: (1) a continuous t-norm with respect to a continuous semicopula is modular if and only if they are equal. The case for a semicopula with respect to a strict t-norm is also the same. A semicopula with respect to a co-semicopula is modular if and only if the semicopula is min and the co-semicopula is max. The modularity condition does not hold for a co-semicopula with respect to a semicopula. (2) Necessary and sufficient conditions are given for a semi-t-operator with respect to a semi-uninorm, a pseudo-uninorm with respect to a semi-t-operator to satisfy the modularity condition equation. New solutions to the modularity condition equations of the Case (1) are characterized.