Finiteness based results in BL-algebras
This paper is a survey of results contained in the works of the authors [Discrete Math. 269, No. 1--3, 93--112 (2003; Zbl 1025.06008)] and of the first author and \textit{P. Niederkorn} [Arch. Math. Logic 44, No. 8, 995--1007 (2005; Zbl 1087.06005)]. \textit{P. Aglianò} and \textit{F. Montagna} [J. Pure Appl. Algebra 181, No. 2--3, 105--129 (2003; Zbl 1034.06009)] have shown that every finite BL-chain is a finite ordinal sum of finite MV-chains. Extending the concept of ordinal sum of BL-algebras, the authors introduce the class of the BL-comets, which is seen as a generalization of the class of the finite BL-algebras. Indeed they prove that any finite BL-algebra is a direct product of BL-comets, so extending the related result for MV-algebras (see, for instance, the book [\textit{R. L. O. Cignoli, I. M. L. D'Ottaviano} and \textit{D. Mundici}, Algebraic foundations of many-valued reasoning. Dordrecht: Kluwer Academic Publishers (2000; Zbl 0937.06009)]). Afterwards, the authors consider the varietes generated by one non-trivial finite BL-chain and they show the existence of a strong duality for these varieties, so obtaining extensions of results of \textit{B. A. Davey} [Trans. Am. Math. Soc. 221, 119--146 (1976; Zbl 0319.06007)] for varieties generated by one finite Heyting chain, and of \textit{P. Niederkorn} [J. Math. Anal. Appl. 255, No. 1, 58--73 (2001; Zbl 0974.06006)] for varieties generated by one finite MV-chain.
scientific article; zbMATH DE number 5009487
scientific article