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

: 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 cl Y : I ( Y ) → I ( Y ) , we deﬁne a map cl ← Y on I ( X ) by A (cid:55)→ ϕ − 1 ( ϕ ( A ) cl Y ) . We prove that, if “ cl Y ” is a weak closure operation (respectively, semi-prime and meet) on I ( Y ) , then so is “ cl ← Y ” on I ( X ) . In addition, for mapping cl X : I ( X ) → I ( X ) , we deﬁne a map cl → X on I ( Y ) as follows: B (cid:55)→ ϕ ( ϕ − 1 ( B ) cl X ) . We show that, if “ cl X ” is a weak closure operation (respectively, semi-prime and meet) on I ( X ) , then so is “ cl → X ” on I


Introduction
Imai and Iseki introduced the concept of BCK-algebra in 1966.BCK-algebra is a generalization of set-theoretic difference.In addition, in [1], Bordbar et al. introduced the concept of weak closure operation as a generalization of closure operation on ideals of a BCK-algebras.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].Moreover, using the notation of weak closure operation, some algebraic structures have been defined and related properties investigated [6][7][8].
In this article, the notions of semi-prime, meet, and prime weak closure operation are introduced, and their relations and properties investigated.Using an epimorphism ϕ : X → Y of lower BCK-semilattices X and Y, mapping cl Y : I(Y) → I(Y), and any ideal A of X with ker(ϕ) ⊆ A, we define a new map cl ← Y on I(X) by and investigate related properties of this map.We show that, if "cl Y " is a weak closure operation (respectively, semi-prime, meet, and prime) on I(Y), then so is "cl ← Y " on I(X).Similarly, using a map cl X : I(X) → I(X) and any ideal B of Y, we define a new map cl → X on I(Y) as follows: B → ϕ(ϕ −1 (B) cl X ).
We prove that, if "cl X " is a weak closure operation (respectively, semi-prime, meet, and prime) on I(X), then so is "cl → X " on I(Y).

Preliminaries
BCI/BCK-algebra, which was introduced by K. Iséki, is a class of logical algebras.In what follows, we gather some necessary definitions and theorems which we need for further results.
An algebra (X; * , 0) of type (2, 0) is called a BCI-algebra if it satisfies the following conditions: Moreover, if an arbitrary BCI-algebra X satisfies the following condition, then X is called a BCK-algebra: Lemma 1. Suppose that X is a BCI/BCK-algebra.Then, the following axioms are satisfied: We define ≤ such that x * y = 0. if and only if x ≤ y, and call it BCK-order.
For a BCK-algebra X, if X concerning the BCK-order is a lower semilattice, then we called it the lower BCK-semilattice.Definition 1. Suppose that X is a BCK-algebra and A is a subset of X.A is called an ideal of X when the following assertions hold: The following condition is satisfied for any ideal A of a BCK-algebra X Remark 1.Let A be a subset of X.The intersection of all ideals of X containing A is an ideal of X and it is denoted by A .Moreover, If A is a finite set, then we say that A is finitely generated ideal of X.
From now on, X and I(X) are the lower BCK-semilattice and the set of all ideals of the X, respectively, unless otherwise specified.In addition, for more information regarding the ordered algebra and BCI/BCK-algebras, please refer to the books [9,10].

Definition 2 ([1]
). Suppose that X is a lower BCK-semilattice and x ∈ X.Then, the element x is called a zeromeet element of X if the following condition holds.
The set of all zeromeet elements of X is denoted by Z(X) and defined as follow: It is clear that 0 ∈ Z(X) and also, for the greatest element 1, we have 1 ∈ X \ Z(X).

Definition 3 ([6]
).Let A and B be nonempty subsets of X; define A ∧ B as follows: and call it the meet ideal of X generated by A and B. Furthermore, {a} ∧ B is denoted by a ∧ B and, similarly, A ∧ {b} is denoted by A ∧ b.
Theorem 1 ([6]).If A and B are ideals of X, then the set

Definition 4 ([8]
). Define a set (A : ∧ B) as follow for nonempty subsets A and B of X and call it the relative annihilator of B with respect to A.
).The relative annihilator (A : ∧ B) of B with respect to A is an ideal of X, where A and B are ideals of X.

Lemma 4 ([8]
).For ideal A of X, it is clear that (A : ∧ X) = A and (A : ∧ A) = X.Now, we define a weak closure operation.

Definition 5 ([1]
).A weak closure operation on set of ideals of a BCK-algebra is defined as a mapping cl : I(X) → I(X), if the following conditions hold.
For a weak closure operation cl, ff the condition holds, then we call it a closure operation on I(X) (see [3]).
For simplicity, we write A cl instead of cl(A).

Definition 7 ([5]
).A meet weak closure operation "cl" on I(X) is defined as follows: Definition 8.A prime weak closure operation "cl" on I(X) is a semi-prime and meet weak closure operation on I(X).
The notions of semi-prime map and meet map are independent.
The following examples show it.
Proof.Let G and H be ideals of Y. Assume that ϕ : and thus there exist g ∈ G and h ∈ H cl Y such that ϕ(z) = g ∧ h.Since ϕ is an isomorphism, we have Proof.Since A and B are ideals, we have A ∧ X B = {a ∧ X b | a ∈ A and b ∈ B}.In addition, since ϕ is epimorphism, for every a ∈ A and b ∈ B by using Lemma 6, we have ϕ(a Theorem 3. Let ϕ : X → Y be an epimorphism of lower BCK-semilattices.Given a mapping cl Y : I(Y) → I(Y) and any ideal A of X with ker(ϕ) ⊆ A, define a map cl ← Y on I(X) as follows: (1) If "cl Y " is a weak closure operation on I(Y), then "cl ← Y " is a weak closure operation on I(X).
(2) If "cl Y " is a semi-prime map on I(Y), then "cl ← Y " is a semi-prime map on I(X).
(3) If "cl Y " is a meet map on I(Y) and ϕ is an isomorphism, then "cl ← Y " is a meet map on I(X).
Proof.(1) Suppose that "cl Y " is a weak closure operation on I(Y).Let A be an ideal of X.Then, ϕ(A) is an ideal of Y by Lemma 7, and thus ϕ(A) ⊆ ϕ(A) cl Y .It follows from Lemma 8 that Now, let A and B be ideals of X such that A ⊆ B. Then, ϕ(A) ⊆ ϕ(B), and thus Therefore, "cl ← Y " is a weak closure operation on I(X).(2) Suppose that "cl Y " is a semi-prime map on I(Y).For any ideals A and B of X, we have by Lemma 6, which implies from Lemmas 8 and 9 that Therefore, "cl ← Y " is a semi-prime map on I(X).
(3) Suppose that "cl Y " is a meet map on I(Y) and ϕ is an isomorphism.For any ideal A of X and z ∈ X \ Z(X), we know that ϕ(z) is a non-zeromeet element of Y by Theorem 2. Using Lemmas 6, 8, and 10, we have Therefore, "cl ← Y " is a meet con I(X).
Corollary 1.If ϕ : X → Y is an isomorphism of lower BCK-semilattices and "cl Y " is a prime weak closure operation on I(Y), then "cl ← Y " is a prime weak closure operation on I(X).
Proof.The proof is straightforward.
Theorem 4. Let ϕ : X → Y be an epimorphism of lower BCK-semilattices.Given a mapping cl X : I(X) → I(X) and any ideal B of Y, define a map cl → X on I(Y) as follows: (1) If "cl X " is a weak closure operation on I(X), then "cl → X " is a weak closure operation on I(Y).
(2) If "cl X " is a semi-prime map on I(X), then "cl → X " is a semi-prime map on I(Y).
(3) If "cl X " is a meet map on I(X) and ϕ is an isomorphism, then "cl → X " is a meet map on I(Y).
Proof.(1) Suppose that "cl X " is a weak closure operation on I(X) and let A and B be ideals of Y.
(2) Let "cl X " be a semi-prime map on I(X) and let A and B be ideals of Y.Then, we have In addition, other kinds of closure operations such as meet, tender, naive, etc. can be checked for homomorphic image and inverse image.For further information about other kinds of (weak) closure operation, please refer to [3,4].Beside, for future research, we invite the researchers to join us and apply the results of this paper to concepts in [11][12][13] about the heigh of ideals.More precisely, we mean what happens for the height of ideals of BCK-algebra under homomorphisms.