Fundamental Journal of Mathematics and Applications, 5 (1) (2022) 63-66

Research Article

Fundamental Journal of Mathematics and Applications

Journal Homepage: www.dergipark.org.tr/en/pub/fujma

ISSN: 2645-8845

doi: https://dx.doi.org/10.33401/fujma.1002715

Exact Sequences of BCK-Modules
Alper Ülker

Department of Mathematics and Computer Science, Istanbul Kültür University, 34256, Istanbul, Turkey

Article Info

Keywords: BCK-algebra, BCK-module,
Exact sequence, Hom functor
2010 AMS: 06F35, 06F25
Received: 30 September 2021
Accepted: 15 February 2022
Available online: 23 February 2022

Abstract

BCK-modules were introduced as an action of a BCK-algebra over an Abelian group.
Homomorphisms of BCK-modules form an exact sequence which is called BCK-sequence.
In this paper, we study homomorphisms of BCK-modules. We show that this homomor-
phisms have a module structure. Moreover, we show that sequences of Hom functors are
BCK-sequences.

1. Introduction

BCK/BCI-algebras were introduced by Imai and Iseki [1, 2]. BCK/BCI-algebras have been studied by many authors, extensively.
In 1994, the BCK-module structure of BCK-algebras was introduced as an action on an Abelian group [3]. In [4], exact
sequences of BCK-modules were studied. Further, in [5],the authors studied the homomorphisms between BCK-modules
and they showed that the set of homomorphisms of BCK-modules form a BCK-module. Later, in [6], homology theory of
BCK-modules was investigated. In [7], the authors studied BCK-sequences and finitely presented BCK-modules.
The paper organized as follows; in section 2, we give general theory of BCK-algebras and BCK-modules. In section 3, we
study the exactness of modules of homomorphisms between BCK-modules.

2. Preliminaries

In this section we introduce the background informations about BCK-algebras, BCK-modules and X-homomorphisms.

Definition 2.1. [8] A BCK-algebra is an algebra (X ;∗,0) of type (2,0) which satisfies the following axioms:
for all p,q,r ∈ X,

1. ((p∗q)∗ (p∗ r))∗ (r ∗q) = 0,
2. (p∗ (p∗q))∗q = 0,
3. (p∗ p) = 0,
4. p∗q = 0 = q∗ p implies p = q.
5. 0∗ p = 0.

Moreover, the relation ≤ can be defined as p≤ q if and only if p∗q = 0, for any p,q ∈ X , is a partial-order on X which is
called BCK-ordering of X .

Definition 2.2. [6] Let (X ;∗,0) be a BCK-algebra and M be an Abelian group under addition +, then M is said to be an
(left) X-module, if there is a mapping (x,m) 7→ xm from X ×M→ M such that it satisfies the following conditions for all
x,x1,x2 ∈ X and m,m1,m2 ∈M:

1. (x1∧ x2)m = x1(x2m),

Email address and ORCID number: a.ulker@iku.edu.tr, 0000-0001-5592-7450

https://orcid.org/0000-0001-5592-7450


64 Fundamental Journal of Mathematics and Applications

2. x(m1 +m2) = xm1 + xm2,
3. 0m = 0

where, x1∧ x2 = x2 ∗ (x2 ∗ x1). If X is bounded with maximal element 1, then

4. 1m = m.

The right X-module can be defined similarly. This X-module M is an BCK-module. If a subgroup N of the X-module M is
also an X-module, then N is called a submodule.
Let M and N be X-modules. A mapping φ : M→ N is said to be an X-homomorphism, if for any x ∈ X and m1,m2 ∈M the
followings hold:

1. φ(m1 +m2) = φ(m1)+φ(m2),
2. φ(xm1) = xφ(m1).

If φ is both injective and surjective, then φ is an X-isomorphism. We say M is isomorphic to N if φ is an X-isomorphism and
denote it by M ∼= N.
The bounded implicative BCK-algebras form a BCK-module over itself (Abujabal et al., 1994). This section devoted to the
examples of BCK-modules.

Example 2.3. Let (X ;∗,0) be a bounded implicative BCK-algebra with X = {0,x,y,1}. Let M = {0,x} be a subset of X. If we
define addition operation + as x+ y = (x∗ y)∨ (y∗ x) and xm = x∧m for all x ∈ X, m ∈M, then M is an X-module. Cayley
table of these operations are as follows:

* 0 x y 1
0 0 0 0 0
x x 0 x 0
y y y 0 0
1 1 y x 0

+ 0 x
0 0 x
x x 0

∧ 0 x
0 0 0
x 0 x
y 0 0
1 0 x

3. Exact BCK-sequences

Definition 3.1. [7] The sequence of X-module homomorphisms M1
f−→M2

g−→M3 is said to be exact at M2, if Im( f ) = Ker(g). A

sequence of X-module homomorphisms, M1
f1−→M2

f2−→ . . .
fn−1−−→Mn is called exact sequence of X-modules, if Im( fi)=Ker( fi+1)

for all i ∈ {1,2, ...,n}.

Theorem 3.2. Let X be a BCK-algebra and K,L and M be X-modules. If A is an X-module and 0→ K
ψ−→ L

φ−→M is exact,
then

0→ Hom(A,K)
ψ∗−→ Hom(A,L)

φ∗−→ Hom(A,M)

is an exact sequence of X-modules.

Proof. First we show that ψ∗ is a monomorphism. Let θ : A→ K be a X-homomorphism with ψ∗θ = 0. Since ψ is a
monomorphism, then for any a ∈ A, the identity ψ∗θ(a) = 0 implies that θ(a) = 0. Thus θ = 0. Hence ψ∗ is a monomorphism.
Let b ∈ Im(ψ∗)⊆ Hom(A,L). Then there exists a ∈ Hom(A,K) such that ψ∗(a) = b = ψa. Since φ∗(b) = φ∗(ψa) = φψa =
0a = 0, we have b∈Ker(φ∗). Hence Im(ψ∗)⊆Ker(φ∗). Let u∈Ker(φ∗)⊆Hom(A,L). Then φ∗(u) = 0 and φu(a) = 0 for any
a ∈ A. The exactness of the sequence gives that Ker(φ) = ψ(K). Thus there exists an x ∈ K which satisfies ψ(x) = u(a). Then
v(a) = x defines a homomorphism v : A→ K with ψ∗(v) = u. Thus Ker(φ∗)⊆ Im(ψ∗). Therefore Ker(φ∗) = Im(ψ∗).

Theorem 3.3. Let X be a BCK-algebra and K,L and M be X-modules. If A is an X-module and K
ψ−→ L

φ−→M→ 0 is exact,
then



Fundamental Journal of Mathematics and Applications 65

0→ Hom(M,A)
φ∗−→ Hom(L,A)

ψ∗−→ Hom(K,A)

is an exact sequence of X-modules.

Proof. First we show that φ∗ is a monomorphism. Let θ : M→ A be an X-homomorphism and θ ∈ Ker(φ∗). Since 0 = φ∗θ =
θφ , this implies that θ(φ(l)) = 0 for all l ∈ L. Thus θ(m) = 0 for all m ∈ Im(φ). The fact that φ is epimorphism implies that
Im(φ) = M and θ = 0. Hence φ∗ is a monomorphism.
Let b ∈ Im(φ∗) ⊆ Hom(L,A). Then there exists a ∈ Hom(M,A) such that φ∗(a) = b = aφ . Since ψ∗(b) = ψ∗(aφ) and
ψ∗(aφ) = aφψ = a0 = 0, this implies that b ∈ Ker(ψ∗). Hence Im(φ∗) ⊆ Ker(ψ∗). Let u ∈ Ker(ψ∗) ⊆ Hom(L,A). Then
ψ∗(u) = 0 = uψ . Following the diagram,

There exists p ∈ Hom(M,A) such that u = pφ = φ∗(p). This implies that u ∈ Im(φ∗). Thus Ker(ψ∗) ⊆ Im(φ∗). Therefore
Ker(ψ∗) = Im(φ∗).

Definition 3.4. Let X be a BCK-algebra and M,N and K be X-modules. If the following sequence of X-modules is exact. Then

0→M→ N→ K→ 0

is called short exact sequence.

Theorem 3.5. Let X be a BCK-algebra and M,N and K be X-modules. If the short sequence of X-homomorphisms is exact;

then followings are equivalent;

1. There exists an X-homomorphism η : N→M such that ηψ = 1M .
2. Submodule Im(ψ) is a direct summand of N.
3. There exists an X-homomorphism θ : K→ N suct that φθ = 1K .

Moreover, we have N ∼= M⊕K.

Proof. 1⇒ 2 Let x ∈ N be any element. Since η(x−ψη(x)) = η(x)− ((ηψ)η(x)) = η(x)− η(x) = 0, then we have
x−ψη(x) ∈ Ker(η). This implies that x = ψ(η(x))+(x−ψη(x)) ∈ Im(ψ)+Ker(η).
Let ψ(m) ∈ Im(ψ)∩Ker(η). Since m = ηψ(m) = η(ψ(m)) = 0, one can conclude that Im(ψ)∩Ker(η) = 0. Hence
N = Im(ψ)⊕Ker(η).
2⇒ 3 Let N′ be a submodule of N and N = Im(ψ)⊕N′. Now since N′∩Ker(φ)=N′∩Im(ψ)= 0, the φ |N′ is a monomorphism.
The fact that φ is a epimorphism implies that there exists x in N for every y ∈ K such that φ(x) = y. If we set x = ψ(a)+b
for a ∈M,b ∈ N′. Then y = φ(x) = φ(ψ(a)+b) = φψ(a)+φ(b) = φ(b). This implies that φ |N′ is an epimorphism. Thus
φ |N′ is an isomorphism. Since φ |N′ is an isomorphism, we can conclude that φ |N′ has an inverse (φ |N′)−1 : K → N for
θ := (φ |N′)−1 : K→ N then we have φθ = 1K .
3⇒ 1 Since φ(n−θφ(n)) = φ(n)−φ(θφ(n)) = 0, we have n−θφ(n) ∈Ker(φ) = Im(ψ). Then there exists m ∈M such that
ψ(m) = n−θφ(n). This m is unique, since ψ is a monomorphism. Set η : N→M and η(n) = m with η is a homomorphism.
The equality,

ψ(m)−θφ(ψ(m)) = ψ(m)−θ(φψ(m)) = ψ(m)−θ(0) = ψ(m), for every m in M.

holds, since φψ(n) = 0. It follows that ψ(m) = ψ(m)−θφ(ψ(m)), and combining this equality with ψ(m) = n−θφ(n),
we can deduce that ψ(m) = n. Thus η(ψ(m)) = m, so we have ηψ = 1M . Since ψ is a monomorphism, then Im(ψ) ∼= M.
Therefore, N ∼= M⊕K.

Acknowledgements

The authors would like to express their sincere thanks to the editor and the anonymous reviewers for their helpful comments
and suggestions.



66 Fundamental Journal of Mathematics and Applications

Funding

There is no funding for this work.

Availability of data and materials

Not applicable.

Competing interests

The authors declare that they have no competing interests.

Author’s contributions

All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.

References

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[2] K. Iseki, S. Tanaka, An introduction to the theory of BCK-algebras, Math. Japonica, 21 (1978), 351-366.
[3] H. A. S. Abujabal, M. Aslam, A. B. Thaheem, On actions of BCK-algebras on groups, 4 (1994), 43-48.
[4] Z. Perveen, M. Aslam, A. B. Thaheem, On BCK-modules, Southeast Asian Bull. Math., 30 (2006), 317-329.
[5] I. Baig, M. Aslam, On certain BCK-modules, Southeast Asian Bull. Math., 34 (2010), 1-10.
[6] A. Kashif, M. Aslam, Homology theory of BCK-modules, Southeast Asian Bull. Math., 38 (2014), 61-72.
[7] S. M. Seyedjoula, A. Gilani, E. Arfa, Exact seqnece in BCK-algebra, J. Inf. Optim. Sci., 41(4) (2020), 1153-1162.
[8] J. Meng, Y. B. Jun, BCK-Algebras, Kyugmoon Sa Co, Korea, 1994.


	Introduction
	Preliminaries
	Exact BCK-sequences