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1

Fuzzy n-fold integral filters in BL-algebras

100%

Borzooei R. , Paad A.

Discussiones Mathematicae - General Algebra and Applications

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2013

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tom 33

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nr 1

57-71

EN

In this paper, we introduce the notion of fuzzy n-fold integral filter in BL-algebras and we state and prove several properties of fuzzy n-fold integral filters. Using a level subset of a fuzzy set in a BL-algebra, we give a characterization of fuzzy n-fold integral filters. Also, we prove that the homomorphic image and preimage of fuzzy n-fold integral filters are also fuzzy n-fold integral filters. Finally, we study the relationship among fuzzy n-fold obstinate filters, fuzzy n-fold integral filters and fuzzy n-fold fantastic filters

2

On implicative and maximal ideals of BL-algebras

100%

Walendziak A.

Commentationes Mathematicae

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2014

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tom Vol 54, No. 2

247--258

EN

In the theory of MV-algebras, implicative ideals are studied by Hoo and Sessa. In this paper we define and characterize implicative ideals of BL-algebras. We also investigate maximal ideals of BL-algebras and prove that if an ideal is prime and implicative, then it is maximal. Moreover, we show that an ideal is maximal if and only if the quotient BL-algebra is a simple MV-algebra. Finally, we give the homomorphic properties of implicative and maximal ideals.

3

Relation between (fuzzy) Gödel ideals and (fuzzy) Boolean ideals in BL-algebras

100%

Paad A.

Discussiones Mathematicae - General Algebra and Applications

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2016

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tom 36

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nr 1

45-58

EN

In this paper, we study relationships between among (fuzzy) Boolean ideals, (fuzzy) Gödel ideals, (fuzzy) implicative filters and (fuzzy) Boolean filters in BL-algebras. In [9], there is an example which shows that a Gödel ideal may not be a Boolean ideal, we show this example is not true and in the following we prove that the notions of (fuzzy) Gödel ideals and (fuzzy) Boolean ideals in BL-algebras coincide.

4

On annihilators in BL-algebras

88%

Zou Y. X. , Xin X. L. , Fei He P.

Open Mathematics

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2016

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tom 14

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nr 1

324-337

EN

In the paper, we introduce the notion of annihilators in BL-algebras and investigate some related properties of them. We get that the ideal lattice (I(L), ⊆) is pseudo-complemented, and for any ideal I, its pseudo-complement is the annihilator I⊥ of I. Also, we define the An (L) to be the set of all annihilators of L, then we have that (An(L); ⋂,∧An(L),⊥,{0}, L) is a Boolean algebra. In addition, we introduce the annihilators of a nonempty subset X of L with respect to an ideal I and study some properties of them. As an application, we show that if I and J are ideals in a BL-algebra L, then [...] JI⊥$J_I^ \bot $ is the relative pseudo-complement of J with respect to I in the ideal lattice (I(L), ⊆). Moreover, we get some properties of the homomorphism image of annihilators, and also give the necessary and sufficient condition of the homomorphism image and the homomorphism pre-image of an annihilator to be an annihilator. Finally, we introduce the notion of α-ideal and give a notation E(I ). We show that (E(I(L)), ∧E, ∨E, E(0), E(L) is a pseudo-complemented lattice, a complete Brouwerian lattice and an algebraic lattice, when L is a BL-chain or a finite product of BL-chains.

5

Interior and closure operators on bounded commutative residuated l-monoids

88%

Rachůnek J. , Švrček F.

Discussiones Mathematicae - General Algebra and Applications

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2008

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tom 28

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nr 1

11-27

EN

Topological Boolean algebras are generalizations of topological spaces defined by means of topological closure and interior operators, respectively. The authors in [14] generalized topological Boolean algebras to closure and interior operators of MV-algebras which are an algebraic counterpart of the Łukasiewicz infinite valued logic. In the paper, these kinds of operators are extended (and investigated) to the wide class of bounded commutative Rl-monoids that contains e.g. the classes of BL-algebras (i.e., algebras of the Hájek's basic fuzzy logic) and Heyting algebras as proper subclasses.

6

A Note on Axiom System for SBL-algebras

75%

Halaš R.

Fundamenta Informaticae

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2009

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tom Vol. 90, nr 1-2

87-92

EN

In the paper we resolve an open problem posed by D.Ciucci in [1] on the independence of axioms for SBL-algebras. The main aim is to present its solution which is, in contrast to the one presented in [4], simple, transparent and easily verifiable.

7

Semitopological BL-algebras and MV-algebras

75%

Zahiri O. , Borzooei R. A.

Demonstratio Mathematica

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2014

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tom Vol. 47, nr 3

522--538

EN

In this paper, by considering the notion of upsets, for any element x of a BL-algebra L, we construct a topology γx on L and show that L-algebras with this topology formes a semitopological BL-algebras. Then we obtain some of the topological aspects of this structure such as connectivity and compactness. Moreover, we introduced two kinds of semitopological MV-algebra by using two kinds of definition of MV–algebra and show that they are equivalent.

8

Folding theory applied to BL-algebras

75%

Jun Y. , Ko J.

Open Mathematics

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2004

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tom 2

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nr 4

584-592

EN

The notion of n-fold grisly deductive systems is introduced. Some conditions for a deductive system to be an n-fold grisly deductive system are provided. Extension property for n-fold grisly deductive system is established.

9

On the lattice of deductive systems of a BL-algebra

63%

Bu§neag D. , Piciu D.

Open Mathematics

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2003

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tom 1

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nr 2

221-237

EN

For a BL-algebra A we denote by Ds(A) the lattice of all deductive systems of A. The aim of this paper is to put in evidence new characterizations for the meet-irreducible elements on Ds(A). Hyperarchimedean BL-algebras, too, are characterized.