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

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

Automated summary

This paper investigates the interval-valued general, residuated, lattice-ordered groupoids and their relevance to IPOFs. It also proposes the notions of expanded interval-valued general, residuated lattice-ordered groupoids and expanded triangle algebras. The one-to-one correspondence between them is explained through a specific proposition. The paper also analyzes the properties of these structures and defines filters, congruence, and quotient structures on the expanded triangle algebras.