The variety generated by perfect BL-algebras: an algebraic approach in a fuzzy logic setting
BL-algebras are the algebras of Hajek's Basic Logic [see \textit{P. Hájek}, Metamathematics of fuzzy logic. Trends in Logic, Studia Logica Library. 4. Dordrecht: Kluwer Academic Publishers (1998; Zbl 0937.03030)]. In the absence of a notion of ideal, the spectral theory of BL-algebras can be investigated using filters -- the latter being the algebraic counterpart of deductive systems. The main results of this paper deal with the variety generated by ``perfect'' BL-algebras, and the class of ``bipartite'' BL-algebras, generalizing previous results on perfect and bipartite MV-algebras. Warning: the authors' usage of the terminology ``locally finite'' (which in the particular case of MV-algebras boils down to ``without homomorphic images'') does not conform to current usage in universal algebra (where \(A\) is ``locally finite'' iff every finitely generated subalgebra of \(A\) is finite). This may lead to confusion: For instance, there exist singly generated infinite MV-subalgebras of \([0,1]\).
scientific article; zbMATH DE number 1761211