Boolean deductive systems of BL-algebras

Boolean deductive systems of BL-algebras,10.1007/s001530100088,Archive for Mathematical Logic,Esko Turunen

Boolean deductive systems of BL-algebras   (Citations: 47)
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.   BL-algebras rise as Lindenbaum algebras from many valued logic introduced by Hjek [2]. In this paper Boolean ds and implicative ds of BL-algebras are defined and studied. The following is proved to be equivalent: (i) a ds D is implicative, (ii) D is Boolean, (iii) L/D is a Boolean algebra. Moreover, a BL-algebra L contains a proper Boolean ds iff L is bipartite. Local BL-algebras, too, are characterized. These results generalize some theorems presented in [4], [5], [6] for MV-algebras which are BL-algebras fulfiling an additional double negation law x = x **.
Journal: Archive for Mathematical Logic - Arch. Math. Log. , vol. 40, no. 6, pp. 467-473, 2001
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    • ...Hajek [13] defined a filter of a BL-algebra A to be a nonvoid subset F of A such that (i) if a; b 2 F then a � b 2 F, and (ii) if a 2 F; a � b, then b 2 F. Note that a subset F of a BL-algebra A is a deductive system of A if and only if F is a filter of A [24]...

    Somayeh MotamedArshamet al. n Fold obstinate filters in BL -algebras

    • ...It is easy to prove that for a subset F A, F is a filter if and only it is a deductive system defined in [9], that is, it satisfies...

    Michiro Kondo. Some Types of Filters in Hoops

    • ...In [7], after introducing Boolean and implicative filters (Boolean deductive systems), Turunen proved in every BL-algebras the equivalence of the above filters...

    Wei Wang. On filters of pseudo MTL-algebras

    • ...Remark 2.7 (Turunen 2001 )I fF is a filter of a BLalgebra A, then we define the equivalence relation x � Fy iff ðx ! y Þ�ð y ! x Þ2 F: Then � F is a congruence and the quotient algebra A/F becomes a BL-algebra with the natural operations induced from those on A. Denoting by x/F the equivalence class of x, then x=F ¼ 1=F iff x 2 F: Conversely, if * is a congruence, then F � : ¼f x 2 A j x � 1g is a filter, and ...

    Lavinia Corina Ciunguet al. State BL-algebras

    • ...Using prime filters of BL-algebras, he proved the completeness of Basic Logic BL. BL-algebras are further discussed by Di Nola ([9] and [10]), Iorgulescu ([19]), Ma([20]) and Turunen ([24], [25]), and so on...
    • ...Recent investigations are concerned with non-commutative generalizations for these structures (see [8, 11 - 17, 22 - 25, 32 - 33])...

    Jianming Zhanaet al. Interval valued -fuzzy filters of pseudo BL -algebras

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