Conditionally distributive uninorms locally internal on the boundary

In the theory of nonadditive integrals, an indispensable step is to define a pair of pseudo-addition and pseudo-multiplication that fulfill the conditional distributivity, leading to a structure of an ordered semiring in some sense. In this paper, we focus on conditionally distributive uninorms locally internal on the boundary, show that the second involved uninorm must be locally internal, and present a general framework of structures of such a pair of uninorms.