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Free algebras in varieties of BL-algebras generated by a BLn-chain

Part of: Algebraic logic General logic Computability and recursion theory Varieties

Published online by Cambridge University Press:  09 April 2009

Manuela Busaniche  and
Roberto Cignoli
Manuela Busaniche
Affiliation:
Departamento de Matemática, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires - CONICET, Ciudad Universitaria, 1428 Buenos Aires, Argentina, e-mail: manuelabusaniche@yahoo.com.ar, cignoli@mate.dm.uba.ar
Roberto Cignoli
Affiliation:
Departamento de Matemática, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires - CONICET, Ciudad Universitaria, 1428 Buenos Aires, Argentina, e-mail: manuelabusaniche@yahoo.com.ar, cignoli@mate.dm.uba.ar
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Abstract

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Free algebras with an arbitrary number of free generators in varieties of BL-algebras generated by one BL-chain that is an ordinal sum of a finite MV-chain Ln, and a generalized BL-chain B are described in terms of weak Boolean products of BL-algebras that are ordinal sums of subalgebras of Ln, and free algebras in the variety of basic hoops generated by B. The Boolean products are taken over the Stone spaces of the Boolean subalgebras of idempotents of free algebras in the variety of MV-algebras generated by Ln.

2000 Mathematics subject classification: primary 03G25, 03B50, 03B52, 03D35, 03G25, 08B20.

Keywords

03G25BL-algebrasresiduated latticeshoopsMoisil algebrasfree algebrasBoolean products.

MSC classification

Secondary: 03G25: Other algebras related to logic 03B50: Many-valued logic 03B52: Fuzzy logic; logic of vagueness 03D35: Undecidability and degrees of sets of sentences 08B20: Free algebras
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 80 , Issue 3 , June 2006 , pp. 419 - 439
Copyright
Copyright © Australian Mathematical Society 2006

References

[1]Aglianò, P., Ferreirim, I. M. A. and Montagna, F., ‘Basic hoops: An algebraic study of continuous t-norms’, manuscript.Google Scholar
[2]Aglianó, P. and Montagna, F., ‘Varieties of BL-algebras I: General properties’, J. Pure Appl. Algebra 181 (2003), 105–129.CrossRefGoogle Scholar
[3]Amer, K., ‘Equationally complete classes of conmutative monoids with monus’, Algebra Universalis 18 (1984), 129–131.CrossRefGoogle Scholar
[4]Balbes, R. and Dwinger, P.. Distributive lattices (University of Missoury Press. Columbia, 1974).Google Scholar
[5]Bigelow, D. and Burris, S., ‘Boolean algebras of factor congruences’, Acta Sci. Math. (Szeged) 54 (1990), 11–20.Google Scholar
[6]Block, W. J. and Ferreinim, I. M. A., ‘Hoops and their implicational reducts (Abstract)’, in: Algebraic Methods in Logic and Computer Sciences, Banach Center Publications 28 (Polish Academy of Science, Warsaw, 1993) pp. 219–230.Google Scholar
[7]Block, W. J. and Ferreinim, I. M. A., ‘On the structure of hoops’, Algebra Universalis 43 (2000), 233–257.CrossRefGoogle Scholar
[8]Boicescu, V., Filipoiu, A., Georgescu, G. and Rudeanu, S., Lukasiewicz-Moisil algebras (Elsevier, Amsterdam, 1991).Google Scholar
[9]Burris, S. and Sankappanavar, H. P., A course in universal algebra (Springer, New York, 1981).CrossRefGoogle Scholar
[10]Busaniche, M., ‘Free algebras in varieties of BL-algebras generated by a chain’, Algebra Universalis 50 (2003), 259–277.CrossRefGoogle Scholar
[11]Cignoli, R., Moisil algebras, Notas de lógica matemática (Instituto De Matemática, Universidad Nac. del Sur, Bahía Blanca, Argentina, 1970).Google Scholar
[12]Cignoli, R., Some algebraic aspects of many-valued logics’, in: Proceedings of the third Brazilian Conference on Mathematical Logic (Sociedade Brasileira de Lógica) (eds. Arruda, A. I., Costa, N. C. A. da and Sette, A. M.) (Sāo Paulo, 1980) pp. 49–69.Google Scholar
[13]Cignoli, R., ‘Proper n-valued łukasiewicz algebras as S-algebras of łukasiewicz n-valued propositional calculi’, Studia Logica 41 (1982), 3–16.CrossRefGoogle Scholar
[14]Cignoli, R., D'Ottaviano, M. I. and Mundici, D., Algebraic foundations of many-valued reasoning (Kluwer, Dordrecht, 2000).CrossRefGoogle Scholar
[15]Cignoli, R. and Torrens, A., ‘An algebraic analysis of product logic’, Mult.- Valued Log. 5 (2000), 45–65.Google Scholar
[16]Cignoli, R. and Torrens, A., ‘Free cancelative hoops’, Algebra Universalis 43 (2000), 213–216.CrossRefGoogle Scholar
[17]Cignoli, R. and Torrens, A., ‘Free algebras in varieties of BL-algebras with a Boolean retract’, Algebra Universalis 48 (2002), 55–79.CrossRefGoogle Scholar
[18]Cignoli, R. and Torrens, A., ‘Hájek basic fuzzy logic and łukasiewicz infinite-valued logic’, Arch. Math. Logic 42 (2003), 361–370.CrossRefGoogle Scholar
[19]Hájek, P., Metamathematics of fuzzy logic (Kluwer, Dordrecht, 1998).CrossRefGoogle Scholar
[20]Horn, A., ‘Free L-algebras’, J. Symbolic Logic 34 (1969), 475–480.CrossRefGoogle Scholar
[21]Iorgulescu, A., ‘Connections between MVn-algebras and n-valued łukasiewicz-moisil algebras Part I’, Discrete Math. 181 (1998), 155–177.CrossRefGoogle Scholar
[22]McNaughton, R., ‘A theorem about infinite-valued sentential logic’, J. Symbolic Logic 16 (1951), 1–13.CrossRefGoogle Scholar
[23]Nola, A. Di, Georgescu, G. and Leustean, L., ‘Boolean products of BL-algebras’, J. Math. Anal. Appl. 251 (2000). 106–131.CrossRefGoogle Scholar
[24]Rodríguez, A. J. and Torrens, A., ‘Wajsberg algebras and post algebras’, Studia Logica 53 (1994), 1–19.CrossRefGoogle Scholar
[25]von Plato, J., ‘Skolem's discovery of Gödel-Dummett logic’, Studia Logica 73 (2003), 153–157.CrossRefGoogle Scholar