Inflationary BL-algebras obtained from 2-dimensional general overlap functions

• Published in: Fuzzy sets and systems Vol. 418; pp. 64 - 83
• Main Authors: Paiva, Rui, Santiago, Regivan, Bedregal, Benjamín, Rivieccio, Umberto
• Format: Journal Article
• Language: English
• Published: Elsevier B.V 15.08.2021
• Subjects: BL-algebras; General overlap functions; Inflationary BL-algebras; Non-associative residuated lattices; Pseudo-automorphisms; Residuation principle; BL-algebras; Residuation principle; Non-associative residuated lattices; General overlap functions; Pseudo-automorphisms; Inflationary BL-algebras
• ISSN: 0165-0114; 1872-6801
• DOI: 10.1016/j.fss.2020.12.018

naBL-algebras are non-associative generalizations of BL-algebras obtained from non-associative t-norms (nat-norms). In the present paper we propose a further generalization of BL-algebras where associativity is not required. Such generalization is based on a subclass of bivariate general overlap functions called inflationary. We call this non-associative generalization inflationary BL-algebras, and we discuss the main differences between the latter and the more specific class of inflationary BL-algebras. We show that the class naBL of non-associative BL-algebras obtained from general overlap functions contains the class naT of naBL-algebras obtained by nat-norms, and we provide a pictorial representation that summarizes these facts. We also prove some related properties, as well as a version of the well-known Chinese Remainder Theorem for these algebras but, under certain restrictions. Moreover, the notions of pseudo-automorphisms, automorphisms and their action on general overlap functions are used to obtain conjugated inflationary BL-algebras, as well as to obtain inflationary BL-algebras by distorting nat-norms by pseudo-automorphisms and, in the converse direction, to obtain naBL-algebras from inflationary BL-algebras via automorphisms.