Pseudo General Overlap Functions and Weak Inflationary Pseudo BL-Algebras

Abstract

General overlap functions are generalized on the basis of overlap functions, which have better application effects in classification problems, and the (weak) inflationary BL-algebras as the related algebraic structure were also studied. However, general overlap functions are a class of aggregation operators, and their commutativity puts certain restrictions on them. In this article, we first propose the notion of pseudo general overlap functions as a non-commutative generalization of general overlap functions, so as to extend their application range, then illustrate their relationship with several other commonly used aggregation functions, and characterize some construction methods. Secondly, the residuated implications induced by inflationary pseudo general overlap functions are discussed, and some examples are given. Then, on this basis, we show the definitions of inflationary pseudo general residuated lattices (IPGRLs) and weak inflationary pseudo BL-algebras, and explain that the weak inflationary pseudo BL-algebras can be gained by the inflationary pseudo general overlap functions. Moreover, they are more extensive algebraic structures, thus enriching the content of existing non-classical logical algebra. Finally, their related properties and their relations with some algebraic structures such as non-commutative residuated lattice-ordered groupoids are investigated. The legend reveals IPGRLs include all non-commutative algebraic structures involved in the article.