On the lattice of deductive systems of a BL-algebra
type: Item
instance of: scholarly article
title: On the lattice of deductive systems of a BL-algebra
zbMATH Open document ID: 1040.03047
DOI: 10.2478/BF02476010
author: Dana Piciu
author: Dimitru Buşneag
published in: Central European Journal of Mathematics
publication date: 2003-01-01T00:00:00Z
review text: 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.
reviewed by: Sergiu Rudeanu
Mathematics Subject Classification ID: 03G25
zbMATH DE Number: 1978734
zbMATH Keywords: BL-algebra
zbMATH Keywords: Boolean algebra
zbMATH Keywords: Archimedean BL-algebra
zbMATH Keywords: hyperarchimedean BL-algebra
zbMATH Keywords: deductive system
zbMATH Keywords: irreducible element
zbMATH Keywords: prime deductive system
zbMATH Keywords: maximal deductive system
MaRDI profile type: Publication
cites work: Q4083730
cites work: Q3159281
cites work: Q4798562
cites work: Algebraic foundations of many-valued reasoning
cites work: Q5519895
cites work: Q4165426
cites work: Q4792046
cites work: Metamathematics of fuzzy logic
cites work: Iséki algebras. Connection with BL algebras
cites work: Q4804773
cites work: An algebraic approach to non-classical logics
cites work: Mathematics behind fuzzy logic
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description: scientific article; zbMATH DE number 1978734