Fuzzy Sets and Systems ( IF 3.2 ) Pub Date : 2020-12-28 , DOI: 10.1016/j.fss.2020.12.018 Rui Paiva , Regivan Santiago , Benjamín Bedregal , Umberto Rivieccio
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 of non-associative BL-algebras obtained from general overlap functions contains the class 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.