Some further construction methods for uninorms via uninorms

In this paper, we further investigate new construction methods for uninorms on bounded lattices via given uninorms. More specifically, we first construct new uninorms on arbitrary bounded lattices by extending a given uninorm on a subinterval of the lattices under necessary and sufficient conditions on the given uninorm. Moreover, based on the new uninorms, we can give another sufficient and necessary condition under which $S_{1}^{*}$ is a $t$-conorm ($T_{1}^{*}$ is a $t$-norm) in \cite{SS06}. Furthermore, using closure operators (interior operators), we also provide new construction methods for uninorms by extending the given uninorm on a subinterval of a bounded lattice under some additional constraints and simultaneously investigate the additional constraints carefully and systematically. Meanwhile, some illustrative examples for the above construction methods of uninorms on bounded lattices are provided.