Finite BL-algebras (Q1402070)

The notion of a BL-algebra was defined and studied by \textit{P. Hájek} [Metamathematics of fuzzy logic. Kluwer, Dordrecht (1998; Zbl 0937.03030)]. A BL-algebra is an algebraic structure \(\mathcal A=(A;\wedge, \vee,\odot, \rightarrow, 0,1)\) of type \((2,2,2,2,0,0)\) satisfying certain axioms; from these it follows that \((A;\wedge,\vee, 0,1)\) is a lattice with the least element 0 and the greatest element 1. If this lattice is a chain, then \(\mathcal A\) is said to be a BL-chain. The authors introduce the notion of a BL-comet, which is a generalization of the concept of a finite BL-chain. The main result of the paper is as follows: Each finite BL-algebra can be represented as a direct product of BL-comets. This is a generalization of the well-known result on MV-algebras saying that each finite MV-algebra is a direct product of MV-chains. It is also proved that there exists a bijection between a class of labelled trees and the class of all finite BL-algebras.