On the lattice of deductive systems of a BL-algebra
The BL-algebras were introduced by \textit{P. Hájek} [Metamathematics of fuzzy logic. Dordrecht: Kluwer (1998; Zbl 0937.03030)] as the algebraic counterpart of ``basic logic''. A deductive system (ds for short) of a BL-algebra \((A;\wedge,\vee,\oplus,\rightarrow,0,1)\) is a subset \(D\subseteq A\) such that \(1\in D\) and if \(x,x\rightarrow y\in D\), then \(y\in D\). A proper ds \(D\) is called prime if \(x\vee y\in D\) implies \(x\in D\) or \(y\in D\). The prime ds coincide with the meet-irreducible elements of the algebraic lattice Ds\((A)\) of all the ds of \(A\). Various other characterizations of prime ds are provided. If every ds has a unique representation as an intersection of prime ds, then Ds\((A)\) is a Boolean algebra. A condition equivalent to the latter property is also given. The last section introduces the notions of archimedean and hyperarchimedean BL algebra and proves that the following conditions are equivalent for a BL-algebra \(A\): (i) \(A\) is hyperarchimedean; (ii) any quotient \(A/D\), where \(D\in \text{Ds}(A)\), is archimedean; (iii) prime ds coincide with maximal ds; (iv) any prime ds is minimal prime.
scientific article; zbMATH DE number 1978734