Abstract
BL-algebras were introduced by P. Hájek as algebraic structures of Basic Logic. The aim of this paper is to survey known results about the structure of finite BL-algebras and natural dualities for varieties of BL-algebras. Extending the notion of ordinal sum of BL-algebras , we characterize a class of finite BL-algebras, actually BL-comets, which can be seen as a generalization of finite BL-chains. Then, just using BL-comets, we can represent any finite BL-algebra A as a direct product of BL-comets. This result can be seen as a generalization of the representation of finite MV-algebras as a direct product of MV-chains. Then we consider the varieties generated by one finite non-trivial totally ordered BL-algebra. For each of these varieties, we show the existence of a strong duality. As an application of the dualities, the injective and the weak injective members of these classes are described.
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References
Agliano P, Montagna F (2003) Varieties of BL-Algebras, I: General properties. J Pure Appl Algebra 181, (2–3):105–129
Balbes R, Dwinger P (1974) Distributive lattices. University of Missouri Press, St. Louis
Birkhoff G (1984) Lattice Theory. American Mathematical Society, Providence
Chang CC (1958) Algebraic analysis of many valued logics. Trans Amer Math Soc 88:467–490
Chang CC (1959) A new proof of the completeness of the Lukasiewicz axioms. Trans Am Math Soc 93:74–80
Cignoli RLO, D'ottaviano IML, Mundici D (2000) Algebraic Foundations of Many-valued Reasoning. (Trends in Logic, Studia Logica Library) Kluwer, Dordrecht
Clark DM, Davey BA (1998) Natural dualities for the working algebraist. Cambridge University Computing, Cambridge
Davey BA (1976) Dualities for Equational Classes of Brouwerian Algebras and Heyting Algebras. Trans Am Math Soc 221: 119–146
Davey BA (1979) On the lattice of Subvarieties. Houston Math J 5:183–192
Davey BA, Werner H (1983) Dualities and equivalences for varieties of algebras. In: Colloquia mathematica societatis János Bolyai, Vol. 33, North-Holland
Di Nola A, Lettieri A (2003) Finite BL-algebras. Discrete Math 269:93–112
Di Nola A, Niederkorn P Natural dualities for varieties of BL-algebras (submitted)
Hájek P (1998) Metamathematics of Fuzzy Logic. (Trends in Logic, Studia Logica Library) Kluwer, Dordrecht
Niederkorn P (2001) Dualities for Varieties of MV-Algebra I. J Math Anal Appl 255:58–73. doi:10.1006/jmaa 2000.7153
Taylor W (1972) Residually small varieties. Alg Universalis 2:33–53
Turunen E (1999) BL-algebras of basic fuzzy logic. Math Soft Comput 6:49–61
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Nola, A., Lettieri, A. Finiteness based results in BL-algebras. Soft Comput 9, 889–896 (2005). https://doi.org/10.1007/s00500-004-0447-7
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DOI: https://doi.org/10.1007/s00500-004-0447-7