# zbMATH — the first resource for mathematics

The prime and maximal spectra and the reticulation of BL-algebras. (English) Zbl 1039.03052
BL-algebras are the algebras of Hájek’s Basic Logic [see P. Hájek, Metamathematics of fuzzy logic. Trends in Logic – Studia Logica Library 4, Kluwer Academic Publishers, Dordrecht (1998; Zbl 0937.03030)]. In the paper under review the author investigates the prime and the maximal spectrum of BL algebras. Equipped with the usual hull-kernel topology, the former turns out to be a spectral space, and the latter a compact Hausdorff space. In the final part of the paper the authors gives the BL-algebraic counterpart of the reticulation functor, along the lines of the Simmons functor (for commutative rings) and of the Belluce functor (for noncommutative rings and MV-algebras). The reticulation functor sends every BL algebra $$A$$ into a normal completely normal lattice $$B$$ in such a way that the prime spectra of $$A$$ and $$B$$ are homeomorphic.

##### MSC:
 03G25 Other algebras related to logic 06D35 MV-algebras 03B52 Fuzzy logic; logic of vagueness
Full Text:
##### References:
 [1] M.F. Atiyah and I.G. Macdonald: Introduction to Commutative Algebra, Addison-Wesley Publishing Company, Reading, Massachussets, Menlo Park, California-London-Don Mills, Ontario, 1969. · Zbl 0175.03601 [2] L.P. Belluce: “Semisimple algebras of infinite valued logic and bold fuzzy set theory”, Can. J. Math., Vol. 38, (1986), pp. 1356-1379. · Zbl 0625.03009 [3] L.P. Belluce: “Spectral spaces and non-commutative rings”, Comm. Algebra, Vol. 19, (1991), pp. 1855-1865. · Zbl 0728.16002 [4] W. Cornish: “Normal lattices”, J. Austral. Math. Soc., Vol. 14, (1972), pp. 200-215. · Zbl 0247.06009 [5] A. Di Nola, G. Georgescu, A. Iorgulescu: “Pseudo-BL algebras: Part I”, Mult.-Valued Log., Vol. 8, (2002), pp. 673-714. · Zbl 1028.06007 [6] A. Di Nola, G. Georgescu, A. Iorgulescu; “Pseudo-BL algebras: Part II”, Mult-Valued Log., Vol. 8, (2002), pp. 717-750. · Zbl 1028.06008 [7] A. Di Nola, G. Georgescu, L. Leuštean: “Boolean products of BL-algebras”, J. Math. Anal. Appl., Vol. 251, (2000), pp. 106-131. http://dx.doi.org/10.1006/jmaa.2000.7024 [8] G. Georgescu: “The reticulation of a quantale”, Rev. Roum. Math. Pures Appl., Vol. 40, (1995), pp. 619-631. · Zbl 0858.06007 [9] G. Grätzer: Lattice Theory. First Concepts and Distributive Lattices, W.H. Freeman and Company, San Francisco, 1972. [10] P. Hájek: Metamathematics of Fuzzy Logic, Trends in Logic-Studia Logica Library 4, Kluwer Academic Publishers, Dordrecht, 1998. [11] M. Mandelker: “Relative annihilators in lattices”, Duke Math. J., Vol. 37, (1970), pp. 377-386. http://dx.doi.org/10.1215/S0012-7094-70-03748-8 · Zbl 0206.29701 [12] A. Monteiro and L’arithm: “etique des filtres et les espaces topologiques. I-II”, Notas de Lógica Mathématica, No. 29-30, Instituto de Mathématica, Univ. Nac. del Sur. Bahia Blanca, Argentina, 1974. · Zbl 0318.06019 [13] K.I. Rosenthal: Quantales and their applications, Longman Scientific and Technical, Longman House, Burnt Mill, 1989. [14] H. Simmons: “Reticulated rings”, J. Algebra, Vol. 66, (1980), pp. 169-192. http://dx.doi.org/10.1016/0021-8693(80)90118-0 · Zbl 0462.13002 [15] E. Turunen: Mathematics behind fuzzy logic, Advances in Soft Computing, Physica-Verlag, Heidelberg, 1999. · Zbl 0940.03029 [16] E. Turunen: “BL-algebras of basic fuzzy logic”, Mathware Soft Comput., Vol. 6, (1999), pp. 49-61. · Zbl 0962.03020 [17] H. Wallman: “Lattices and topological spaces”, Ann. Math. (2), Vol. 39, (1938), pp. 112-126. http://dx.doi.org/10.2307/1968717 · Zbl 0018.33202
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.