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

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 na13f 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 pseudoautomorphisms and, in the converse direction, to obtain naBL-algebras from inflationary BL-algebras via automorphisms. (C) 2020 Elsevier B.V. All rights reserved.

Automated summary

In this paper, a further generalization of BL-algebras without requiring associativity is proposed, based on a subclass of bivariate general overlap functions called inflationary. The differences between non-associative BL-algebras and the more specific class of inflationary BL-algebras are discussed, with a pictorial representation provided to summarize the relationship. Additionally, properties related to these algebras are proven, as well as a version of the Chinese Remainder Theorem under certain restrictions. The concepts of pseudo-automorphisms, automorphisms, and their effects on general overlap functions are utilized to obtain conjugated inflationary BL-algebras and to derive naBL-algebras from inflationary BL-algebras through automorphisms.