Equivalence relations derived from $U$- and $V$-partial orders on bounded lattices: different neutral and zero elements perspective

Abstract

This paper investigates the equivalence relations and algebraic structures induced by $U$-partial and $V$-partial orders on bounded lattices, respectively, with respect to the different neutral and zero elements. The study begins by presenting the equivalence classes associated with two distinct t-norms and their corresponding conjugates. It also explores uninorms and nullnorms where the underlying t-norms and t-conorms are the $\mathrm{N}$-dual/conjugate of each other. The findings indicate that the underlying t-norms and t-conorms are crucial in determining the equivalence classes and the overall structure of these orders. Finally, the relationships between the algebraic structures generated by the orders derived from uninorms and nullnorms are analyzed using two order-preserving bijections. Our work aims to enhance both the theoretical understanding and practical applications of fuzzy logic and ordered algebraic systems.