Axiomatization of non-associative generalisations of Hájek's BL and psBL

• Published in: Journal of applied non-classical logics Vol. 30; no. 1; pp. 1 - 15
• Main Author: Petrukhin, Yaroslav
• Format: Journal Article
• Language: English
• Published: Taylor & Francis 02.01.2020
• Subjects: Mathematical fuzzy logic; naBL-algebras; non-associative logic; non-commutative logic; psnaBL-algebras
• ISSN: 1166-3081; 1958-5780
• DOI: 10.1080/11663081.2019.1703468

In this paper, we consider non-associative generalisations of Hájek's logics BL and psBL. As it was shown by Cignoli, Esteva, Godo, and Torrens, the former is the logic of continuous t-norms and their residua. Botur introduced logic naBL which is the logic of non-associative continuous t-norms and their residua. Thus, naBL can be viewed as a non-associative generalisation of BL. However, Botur has not presented axiomatization of naBL. We fill this gap by constructing an adequate Hilbert-style calculus for naBL. Although, as was shown by Flondor, Georgescu, and Iorgulescu, there are no non-commutative continuous t-norms, Hájek's psBL can be viewed as BL's non-commutative generalisation. We present the logic psnaBL of psnaBL-algebras which can be viewed as naBL's non-commutative generalisation as well as psBL's non-associative generalisation and BL's both non-commutative and non-associative generalisation.