Interval-Valued General Residuated Lattice-Ordered Groupoids and Expanded Triangle Algebras

As an extension of interval-valued pseudo t-norms, pseudo-overlap functions (IPOFs) play a vital role in solving multi-attribute decision making problems. However, their corresponding algebraic structure has not been studied yet. On the other hand, with development non-commutative (non-associative) fuzzy logic, study residuated lattice theory is gradually deepening. Due to conditions operators being weakened, structures are expanding. Therefore, on basis theory, we generalize and research related contents general, residuated, lattice-ordered groupoids. In this paper, concept interval-valued, groupoids given, some examples presented illustrate relevance IPOFs them. Then, order further them, propose notions expanded, expanded triangle algebras, explain that there one-to-one correspondence between them through specific proposition. Some properties also analyzed. Lastly, show definitions filters investigate congruence quotient