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Standard completeness of Hájek basic logic and decompositions of BL-chains. (English) Zbl 1094.03013
Summary: The aim of this paper is to survey the tools needed to prove the standard completeness of Hájek Basic Logic with respect to continuous t-norms. In particular, decompositions of totally ordered BL-algebras into simpler components are considered in some detail.

##### MSC:
 03B52 Fuzzy logic; logic of vagueness 03G25 Other algebras related to logic
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##### References:
 [1] Aglianó P, Montagna F (2003) Varieties of BL-algebras I: general properties, J Pure Appl Alg 181:105-129 · Zbl 1034.06009 [2] Blok WJ, Ferreirim IMA (2000) On the structure of hoops. Algebra Univ 43:233–257 · Zbl 1012.06016 [3] Blok W, Pigozzi D (1989) Algebraizable Logics. Mem Am Math Soc 77(396): · Zbl 0664.03042 [4] Busaniche M, Free algebras in varieties of BL-algebras generated by a chain. Algebra Univ (to appear) · Zbl 1094.03058 [5] Busaniche M (2002) Decomposition of BL-chains. Manuscript [6] Chang CC (1959) A new proof of the completeness of the Lukasiewicz axioms, Trans Am Math Soc 93:74–90 [7] Cignoli R, Mundici D (2001) Partial isomorphisms on totally ordered abelian groups and Hájek’s completeness theorem for basic logic. Mult Valued Log 6:89–94 · Zbl 1020.03019 [8] Cignoli R, D’Ottaviano IM, Mundici D (2000) Algebraic foundations of many-valued reasoning. Kluwer, Dordrecht [9] Cignoli R, Esteva F, Godo L, Torrens A (2000) Basic fuzzy logic is the logic of continuous t-norms and their residua. Soft Comput 4:106–112 [10] Cignoli R, Torrens A (2000) An algebraic analysis of product logic. Mult Valued Log 5:45–65 · Zbl 0962.03059 [11] Cignoli R, Torrens A (2002) Free algebras in varieties of BL-algebras with a Boolean retract. Algebra Univ 48:55–79 · Zbl 1058.03077 [12] Cignoli R, Torrens A (2003) Hájek basic fuzzy logic and Lukasiewicz infinite-valued logic. Arch Math Logic 42:361–370 · Zbl 1025.03018 [13] Esteva F, Godo L, Montagna F, Varieties of BL-algebras generated by continuous t-norms and their residua, Studia Logica (to appear) [14] Hájek P (1998) Metamathematics of fuzzy logic. Kluwer, Dordrecht · Zbl 0937.03030 [15] Hájek P (1998) Basic fuzzy logic and BL-algebras. Soft Comput 2:124–128 [16] Hájek P, Godo L, Esteva F (1996) A complete many-valued logic with product conjunction. Arch Math Logic 35:191–208 · Zbl 0848.03005 [17] Montagna F Generating the variety of BL-algebras (to appear in the present volume) · Zbl 1093.03039 [18] Mostert PS, Shields AL (1957) On the structure of semigroups on a compact manifold with boundary. Ann Math 65:117–143 · Zbl 0096.01203
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