Weak Inflationary BL-Algebras and Filters of Inflationary (Pseudo) General Residuated Lattices

After the research on naBL-algebras gained by the non-associative t-norms and overlap functions, inflationary BL-algebras were also studied as a recent kind of non-associative generalization of BL-algebras, which can be obtained by general overlap functions. In this paper, we show that not every inflationary general overlap function can induce an inflationary BL-algebra by a counterexample and thus propose the new concept of weak inflationary BL-algebras. We prove that each inflationary general overlap function corresponds to a weak inflationary BL-algebra; therefore, two mistaken results in the previous paper are revised. In addition, some properties satisfied by weak inflationary BL-algebras are discussed, and the relationships among some non-classical logic algebras are analyzed. Finally, we establish the theory of filters and quotient algebras of inflationary general residuated lattice (IGRL) and inflationary pseudo-general residuated lattice (IPGRL), and characterize the properties of some kinds of IGRLs and IPGRLs by naBL-filters, (weak) inflationary BL-filters, and weak inflationary pseudo-BL-filters.

Automated summary

After investigating naBL-algebras using non-associative t-norms and overlap functions, the study also extended to inflationary BL-algebras as a recent non-associative generalization of BL-algebras, which can be obtained using general overlap functions. In this paper, it is demonstrated through a counterexample that not every inflationary general overlap function can induce an inflationary BL-algebra, leading to the introduction of the concept of weak inflationary BL-algebras. It is then proven that each inflationary general overlap function corresponds to a weak inflationary BL-algebra, allowing a correction of two mistaken results from a previous paper. Additionally, the properties of weak inflationary BL-algebras are discussed, as well as the relationships among various non-classical logic algebras. Furthermore, the theory of filters and quotient algebras of inflationary general residuated lattices (IGRL) and inflationary pseudo-general residuated lattices (IPGRL) is established, along with the characterization of certain types of IGRLs and IPGRLs using naBL-filters, (weak) inflationary BL-filters, and weak inflationary pseudo-BL-filters.