A Finite Type of Closure Operations on BCK-algebras

Bordbar, Hashem; Zahedi, M. M.

doi:10.18576/amisl/040204

Cited by 8 publications

(8 citation statements)

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“…can be checked for prime ideals in prime decompositions. For further information about other kinds of (weak) closure operation, please refer to [2,3,21,22].…”

Section: Discussionmentioning

confidence: 99%

“…Let X be a BCK-algebra. We defined in [2] the closure operation on I X , as the following function cl :…”

Section: Lemma 8 ([22]mentioning

confidence: 99%

“…For the first time, Aslam et al in [1] discussed the concept of annihilators for a subset in BCK-algebras, and after that many researchers generalized it in different research articles (see [2][3][4][5]). Except these, the notion related to annihilator in BCK-algebras is investigated in the papers [6][7][8].…”

Section: Introductionmentioning

confidence: 99%

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Primeness of Relative Annihilators in BCK-Algebra

2020

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Conditions that are necessary for the relative annihilator in lower B C K -semilattices to be a prime ideal are discussed. Given the minimal prime decomposition of an ideal A, a condition for any prime ideal to be one of the minimal prime factors of A is provided. Homomorphic image and pre-image of the minimal prime decomposition of an ideal are considered. Using a semi-prime closure operation “ c l ”, we show that every minimal prime factor of a c l -closed ideal A is also c l -closed.

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“…can be checked for prime ideals in prime decompositions. For further information about other kinds of (weak) closure operation, please refer to [2,3,21,22].…”

Section: Discussionmentioning

confidence: 99%

“…Let X be a BCK-algebra. We defined in [2] the closure operation on I X , as the following function cl :…”

Section: Lemma 8 ([22]mentioning

confidence: 99%

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Primeness of Relative Annihilators in BCK-Algebra

2020

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“…They investigated some properties of weak closure operations and provided examples. They defined many different types of weak closure operations on ideals of an arbitrary BCK-algebra such that finite type, (strong) semi-primeness, tender, and naive weak closure operations and investigated their related properties [2][3][4][5]. Besides, the notions of commutative, implicative, and positive implicative weak closure operations were investigated by [1].…”

Section: Introductionmentioning

confidence: 99%

Homomorphic Image and Inverse Image of Weak Closure Operations on Ideals of BCK-Algebras

2020

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We introduce the notions of meet, semi-prime, and prime weak closure operations. Using homomorphism of BCK-algebras φ : X → Y , we show that every epimorphic image of a non-zeromeet element is also non-zeromeet and, for mapping c l Y : I ( Y ) → I ( Y ) , we define a map c l Y ← on I ( X ) by A ↦ φ − 1 ( φ ( A ) c l Y ) . We prove that, if “ c l Y ” is a weak closure operation (respectively, semi-prime and meet) on I ( Y ) , then so is “ c l Y ← ” on I ( X ) . In addition, for mapping c l X : I ( X ) → I ( X ) , we define a map c l X → on I ( Y ) as follows: B ↦ φ ( φ − 1 ( B ) c l X ) . We show that, if “ c l X ” is a weak closure operation (respectively, semi-prime and meet) on I ( X ) , then so is “ c l X → ” on I ( Y ) .

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“…Jun et al [14], [2], [3] have discussed soft set theory in residuated lattices. Jun and Park [17], Bordbar [1], [4], [5], [6], [7] and [8] studied applications of soft sets in ideal theory of BCK/BCI-algebras. Jun et al [15,16] introduced the notion of intersectional soft sets, and considered its applications to BCK/BCI-algebras.…”

Section: Introductionmentioning

confidence: 99%