A note on the distributivity for uninorms with noncontinuous underlying operators

Abstract

Ch.Y. Wang, P. Wang and B. Zhang (Fuzzy Sets Syst. 462 (2023) 108403) recently studied the distributivity of a uninorm U1 belonging to two well known classes of uninorms, i.e., Umax (resp. Umin) and Ucos,min (resp. Ucos,max), over a uninorm U2 with a noncontinuous underlying operator. Although they characterized the structure of U1, they just obtained partially the structure of uninorm U2. In this work, we continue their work and try to completely solve the distributivity equation. Specifically, when the first uninorm U1 in the distributivity equation is in Umax (resp. Umin), by letting U2(e1,e1)=k∈[0,e1[ (resp. U2(e1,e1)=k∈]e1,1]), we characterize completely the structure of U1 and U2 under some additional condition. If k=0, then it is the one discussed by Wang et al. When the first uninorm U1 is in Ucos,min (resp. Ucos,max), under the same condition as in Wang et al.'s paper, we completely characterize the structure of U1 and U2.