On an extension of KU-algebras

: In this article we deﬁne an extension of KU-algebra and call it an extended KU-algebra. We study basic properties of this extended KU-algebra and its ideals. We also discuss the relations between extended KU-algebras and KU-algebras.


Introduction
Prabpayak and Leerawat introduced KU-algebras in [9], basic properties of KU-algebras and its ideals are discussed in [9,10].After that many authors widely studied KU-algebras in different directions e.g. in fuzzy, in neutrosophic and in intuitionistic context [17], soft and rough sense etc. Naveed et al. [15] introduced the concept of cubic KU-ideals of KU-algebras whereas Mostafa et al. [7] defined fuzzy ideals of KU-algebras.Further Mostafa et al. [8] studied Interval valued fuzzy KU-ideals in KU-algebras.
Imai and Iseki [14] introduced two classes of abstract algebras namely BCK/BCI algebras as an extension of the concept of set-theoretic difference and proportional calculi.Then onwards many works been done based on this logical algebras.Subrahmanya defined and shown results based on Commutative extended BCK-algebra.Farag and Babiker [5] studied Quasi-ideals and Extensions of BCK-algebras.
Extensions of different algebraic structures whether in classical or logical algebras are intensively studied by many researchers in recent years.Motivated by works based on extension, we have studied an extension of KU-algebras.Some recent work based on extension and generalization of logical algebras can be seen in [11][12][13].
In this article, definitions, examples and basic properties of KU-algebras are given in Section 2. In section 3, extended KU-algebras are defined with examples and related results.In section 4, ideals of extended KU-algebras are studied and section 5 concludes the whole work.

Preliminaries
In this section, we shall give definitions and related terminologies on KU-algebras, KU-subalgebras, KU-ideals with examples and some results based on them.Definition 1. [9] By a KU-algebra we mean an algebra (X, •, 1) of type (2, 0) with a single binary operation • that satisfies the following propoerties: for any x, y, z ∈ X, In what follows, let (X, •, 1) denote a KU-algebra unless otherwise specified.For brevity we also call X a KU-algebra.The element 1 of X is called constant which is the fixed element of X. Partial order " ≤ " in X is denoted by the condition x ≤ y if and only if y In a KU-algebra, the following properties are true: Example 1. [7] Let X = {1, 2, 3, 4, 5} in which • is defined by the following table It is easy to see that X is a KU-algebra.Definition 2. A non-empty subset K of a KU-algebra X is called a KU-ideal of X if it satisfies the following conditions: (1) 1 ∈ K, (2) x ∈ K and x • y ∈ K implies y ∈ K, for all x, y ∈ X.

Extended KU-Algebras
In this section, we give a definition of an extension of KU-algebras and related results.In the whole text by (kue) we mean an extended KU-algebras as defined below.
Definition 3.For a non-empty set X, we define an extended KU-algebra corresponding to a non-empty subset K of X as an algebra (X K ; •, K) such that • is a binary operation on X K satisfies the following axioms: For simplicity we will denote simply X K as an extended KU-algebra (X K , •, K) in the later text.
Example 3. Let X = {1, 2, 3, 4} and K = {1, 2}.Then we can see in the following table that X K is an extended KU-algebra. • Then we can see in the following table that X K is an extended KU-algebra.
Now we have the following properties and basic results of an extended KU-algebra X K .
Theorem 1.Every KU algebra is an extended KU-algebra and converse holds if and only if K is a singleton set.
Conversely, we suppose that an extended KU-algebra X K is a KU-algebra.Take Also, by considering X K as a KU-algebra, we get that . We conclude that k 1 = k 2 = 1 and hence K = {1}.
Lemma 3.Each extended KU-algebra X K , satisfies the following properties for all x, y, z ∈ X : Proof.(i), (ii) and (iii) directly follow from the Definition 4.
Definition 4. We define a binary relation ≤ on an extended KU-algebra X K as, x ≤ y if and only if either x = y or y • x ∈ K and y K.
Note that if y ∈ K and y • x ∈ K for any x ∈ X, then by (kue3) we get, x = y • x ∈ K and x • y = y ∈ K ⇒ x = y.Definition 5. A non-empty subset K of a KU-algebra X is called the minimal set in (X K , ≤) if x ≤ k implies x = k, for any x, y, z ∈ X and k ∈ K. Lemma 4.An extended KU-algebra X K with binary relation ≤ is a partial ordered set with a minimal set K.
Proof.It follows from the definition of ≤ and Lemma 3 (i) that x ≤ x.Let x ≤ y and y ≤ x.If x = y, then we are done, otherwise by the definition of ≤ we get, y • x ∈ K and x • y ∈ K which implies x = y by (kue4).Moreover, if x = y or y = z, then x ≤ z.Otherwise by the definition of ≤ we get, y kue1) and (kue3).
Since x ≤ k ∈ K, therefore it directly follows from the Definition 4 that x = k and hence K is a minimal set.
Taking (X K , ≤) as a partial ordered set we obtain the following properties: Theorem 2. Let X K be an extended KU-algebra with partial order ≤ .Then x and (y • x) • x ≤ y for all x, y, z ∈ X and k ∈ K.
Theorem 3. Let X K 1 and X K 2 be two extended KU-algebras with same operation Similarly we can show that and Y L is also an extended KU-algebra.
The following result derived from the definition of extended KU-algebras.
Proposition 1.If (X i , •, K), for i ∈ Λ, is a family of extended KU-subalgebras of an extended KUalgebra (X K , •, K), then i∈Λ (X i ; •, K) is also an extended KU-subalgebra.
Theorem 4. Let X K be an extended KU-algebra.Then Y L is a sub-algebra of X K if and only if x • y ∈ Y, for all x, y ∈ Y, and L = K ∩ Y.
Proof.Let Y L be a sub-algebra of an extended KU-algebra X K .Then clearly x • y ∈ Y, for all x, y ∈ Y and let M = K ∩ Y. Since M ⊆ K, therefore it is easy to see that Y M is a subalgebra of X K .By Theorem 3, M = L = K ∩ Y. Converse is obvious.
Corollary 1.If X L is a sub-algebra of X K , then L = K.

Ideals on extended KU-algebras
In this section we will discuss ideals and some properties of ideals related to extended KU-algebras.
Definition 7. A subset I of an extended KU-algebra X K is called an ideal of X K if K ⊆ I and x ∈ I, x • y ∈ I ⇒ y ∈ I. Clearly X K itself and K are trivial ideals of X K .
Example 6.In Example 4 we can see that the subset I = {1, 2, 3, 4} is an ideal of the extended KUalgebra X K .
Proposition 2. For any ideal I of extended KU-algebra, X K .If x ∈ I and y ≤ x, then y ∈ I.
Proof.Proof follows from the Definitions 4 and 7.
Proposition 3. Let {I λ : λ ∈ Λ} be a family of ideals of X K .Then λ∈Λ I λ is also ideal of X K .
Proof.Clearly 1 ∈ J. Let x ∈ J and x • y ∈ J for x, y ∈ X .If x = 1, then 1 • y = y ∈ J. Also if x 1 but y = 1, then y ∈ J and we are done.Therefore we suppose that both x, y 1, hence x ∈ I \ K and y ∈ X \ K.If x • y = 1, then by Lemma 3(iii) and (ku3) we get x Example 7. Let X = {a, b, c, d, e} and K = {a, b}.By the following table, X K is an extended KU-algebra.which is a KU-algebra.We can see that I = {a, b, c, d} is an ideal of X K and J = (I \ K) ∪ {1} = {1, c, d} is an ideal of X .Definition 8. We call a map f : (X, If f is an isomorphism, then we say that X K is isomorphic to Y L and write it as, X K Y L .Theorem 6.Let f : (X, • 1 , K) → (Y, • 2 , L) be an isomorphism between two extended KU-algebras.Then f (K) = L.
Proof.Since f is a bijective function and I is an ideal of X K , therefore K ⊆ I and hence f (K) ⊆ f (I).By Theorem 6, f (K) = L ⊆ J = f (I), the rest follows by the fact that f is an isomorphism.

Conclusions
In this paper, an extension for KU-algebras is given as extended KU algebras X K depending on a non-empty subset K of X.We see that every KU-algebra is an extended KU-algebra and extended KUalgebras X K is a KU-algebra X if and only if K is a singleton set.Several properties including extended KU-algebras were explored.We also discuss ideals and isomorphisms related properties on extended KU-algebras.
As a future work one can consider such extensions on other logical algebras.Moreover, several identities such as fuzzification, roughness, codes, soft sets and other related work can be seen on extended KU-algebras.

• a b c d e a a b c d e b a b c d e c a a a a e d a a d a e e a a b a a
Take X = {1, c, d, e} with the following table.