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

Abstract

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.

Overlap Functions-Based Fuzzy Mathematical Morphological Operators and Their Applications in Image Edge Extraction

“…In future work, the mathematical characteristics and extensions of the two groups of fuzzy mathematical morphology operators proposed in this work will be discussed in depth (such as new morphological operators based on pseudo overlap functions and semi-overlap functions, related deep morphological networks, etc.). Moreover, some new results obtained from other papers (see [44][45][46][47]) will be applied to fuzzy mathematical morphology and medical images processing.…”

Pseudo Overlap Functions, Fuzzy Implications and Pseudo Grouping Functions with Applications

“…Therefore, POFs and PGFs have a wide adaptability and application value. There are still many theoretical and applied problems to be studied in terms of pseudo overlap functions and fuzzy logic (see [42][43][44][45][46][47]).…”