The modularity equation for Mayor's aggregation operators and uninorms

The focus of this paper is to investigate the modularity equation involving uninorms and Mayor's aggregation operators.  Necessary and sufficient conditions are established for this equation. And, it finds that themodularity equation of a Mayor's aggregation operator over a uninorm is reduced to the modularityof a commutative semi-t-norm over a uninorm and the modularity equation of a uninorm over a Mayor's aggregation operator is reduced to the modularity equation of a uninorm over a commutative semi-t-conorm. Among them, the cases of a uninorm which is locally internal on the boundary are studied in \cite{Su_Riera_Aguilera_Torrens_۲۰۱۹}. In this paper, we consider whether the neutral element e of the uninorm is idempotent element of the Mayor's aggregation operator in modularity equation to get solutions in the corresponding cases.