Anis et al., Cogent Mathematics (2017), 4: 1352117 https://doi.org/10.1080/23311835.2017.1352117
PURE MATHEMATICS | RESEARCH ARTICLE
Hybrid ideals in semigroups Saima Anis1*, Madad Khan1 and Young Bae Jun2
Received: 04 May 2016 Accepted: 04 July 2017 First Published: 08 July 2017 *Corresponding author: Saima Anis, Department of Mathematics, COMSATS Institute of Information Technology, Abbottabad, Pakistan E-mails:
[email protected],
[email protected] Reviewing editor: Hari M. Srivastava, University of Victoria, Canada Additional information is available at the end of the article
Abstract: The notions of hybrid subsemigroups and hybrid left (resp., right) ideals in semigroups are introduced, and several properties are investigated. Using these notions, characterizations of subsemigroups and left (resp., right) ideals are discussed. The concept of hybrid product is also introduced, and characterizations of hybrid subsemigroups and hybrid left (resp., right) ideals are considered by using the notion of hybrid product. Relations between hybrid intersection and hybrid product are displayed. Subjects: Science; Mathematics & Statistics; Advanced Mathematics; Algebra; Group Theory; Mathematical Logic; Pure Mathematics; Foundations & Theorems Keywords: (characteristic; identity) hybrid structure; hybrid subsemigroup; hybrid left (resp., right) ideal; hybrid product 2010 Mathematics subject classification: 20M12; 20M17; 03E72 1. Introduction The study of the fuzzy algebraic structures has started with the introduction of the concepts of fuzzy (subgroupoids) subgroups and fuzzy (left, fight) ideals in the pioneering paper of Rosenfeld (1971). Since then, several authors applied fuzzy set theory to semigroups, and Mursaleen, Srivastava, and Sunil (2016) studied certain new spaces of statistically convergent and strongly summable sequences of fuzzy numbers. As a parallel circuit of fuzzy sets and soft sets (or, hesitant fuzzy set), Jun, Song, and Muhiuddin (in press) introduced the notion of hybrid structure in a set of parameters over an initial universe set, and applied it to BCK / BCI-algebras and linear spaces. As a new mathematical tool for dealing with uncertainties, Molodtsov (1999) introduced the soft set theory. Torra introduced the concept of a hesitant fuzzy set (Torra, 2010; Torra & Narukawa,
ABOUT THE AUTHORS
PUBLIC INTEREST STATEMENT
Saima Anis and Madad Khan had worked in semigroups and left almost semigroups. Jun introduced the notion of hybrid structure in a set of parameters over an initial universe set. In this paper, we applied hybrid structure to semigroups.
A semigroup is a nonempty set with a binary operation which is associative. An ideal of a semigroup S is a subset A of S such that AS or SA is contain in S. We introduced new kind of ideals in semigroups called as hybrid ideals and have investigated various properties in it. Using these notions, we have considered characterizations of subsemigroups and left (right) ideals. We also have introduced the concept of hybrid product, and have discussed characterizations of hybrid subsemigroups and hybrid left (resp., right) ideals by using the notion of hybrid product. We have provided relations between hybrid intersection and hybrid product. Using the notions and results in this paper, we will study the hybrid structures in related algebraic structures and decision making problems etc.
© 2017 The Author(s). This open access article is distributed under a Creative Commons Attribution (CC-BY) 4.0 license.
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2009) which is a generalization of Zadeh’s fuzzy set (Zadeh, 1965). The hesitant fuzzy set is very useful to express peoples hesitancy in daily life, and it is a very useful tool to deal with uncertainty, which can be accurately and perfectly described in terms of the opinions of decision makers. In this paper, we apply the notion of hybrid structure to semigroups. We introduce the notions of hybrid subsemigroups and hybrid left (resp., right) ideals in semigroups, and investigate several properties. Using these notions, we consider characterizations of subsemigroups and left (resp., right) ideals. We also introduce the concept of hybrid product, and discuss characterizations of hybrid subsemigroups and hybrid left (resp., right) ideals by using the notion of hybrid product. We provide relations between hybrid intersection and hybrid product.
2. Preliminaries 2.1. Fundamentals on semigroups Let L be a semigroup. Let A and B be subsets of L. Then the multiplication of A and B is defined as follows:
{ } AB = ab ∈ L ∣ a ∈ A and b ∈ B . A semigroup L is said to be regular if for every x ∈ L there exists a ∈ L such that xax = x. A nonempty subset A of L is called • a subsemigroup of L if AA ⊆ A, i.e. ab ∈ A for all a, b ∈ A, • a left (resp., right) ideal of L if LA ⊆ A (resp. AL ⊆ A), i.e. xa ∈ A (resp. ax ∈ A) for all x ∈ L and
a ∈ A. • a two-sided ideal of L if it is both a left and a right ideal of L.
2.2. Fundamentals on hybrid structures In what follows, let I be the unit interval, L a set of parameters and 𝒫(U) denote the power set of an initial universe set U. Definition 2.1 (Jun et al., in press) A hybrid structure in L over U is defined to be a mapping ( ) f̃𝜆 : = (f̃ , 𝜆):L → 𝒫(U) × I, x ↦ f̃ (x), 𝜆(x)
where f̃ :L → 𝒫(U) and 𝜆:L → I are mappings. Let us denote by H(L) the set of all hybrid structures in L over U. We define an order ≪ in H(L) as follows:
( )( ) ̃ g, ̃ 𝜆⪰𝛾 ∀f̃𝜆 , g̃ 𝛾 ∈ H(L) f̃𝜆 ≪ g̃ 𝛾 ⇔ f̃ ⊆
(2.1)
̃ g̃ means that f̃ (x) ⊆ g(x) ̃ and 𝜆 ⪰ 𝛾 means that 𝜆(x) ≥ 𝛾(x) for all x ∈ L. Note that where f̃ ⊆ (H(L), ≪) is a poset. Definition 2.2 (Jun et al., in press) Let f̃𝜆 be a hybrid structure in L over U. Then the sets { f̃𝜆 [𝛼, t]: = x ∈ X| f̃ (x) ⊇ 𝛼, { f̃𝜆 (𝛼, t]: = x ∈ X| f̃ (x) ⊋ 𝛼, { f̃𝜆 [𝛼, t): = x ∈ X| f̃ (x) ⊇ 𝛼, { f̃𝜆 (𝛼, t): = x ∈ X| f̃ (x) ⊋ 𝛼,
} 𝜆(x) ≤ t , } 𝜆(x) ≤ t , } 𝜆(x) < t , } 𝜆(x) < t
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are called the [𝛼, t]-hybrid cut, (𝛼, t]-hybrid cut, [𝛼, t)-hybrid cut, and (𝛼, t)-hybrid cut of f̃𝜆, respectively, where 𝛼 ∈ 𝒫(U) and t ∈ I. Obviously, f̃𝜆 (𝛼, t) ⊆ f̃𝜆 (𝛼, t] ⊆ f̃𝜆 [𝛼, t] and f̃𝜆 (𝛼, t) ⊆ f̃𝜆 [𝛼, t) ⊆ f̃𝜆 [𝛼, t].
Definition 2.3 (Jun et al., in press) Let f̃𝜆 and g̃ 𝛾 be hybrid structures in L over U. Then the hybrid intersection of f̃𝜆 and g̃ 𝛾 is denoted by f̃𝜆 ⋒ g̃ 𝛾 and is defined to be a hybrid structure ( ) ̃ (𝜆 ∨ 𝛾)(x) , f̃𝜆 ⋒ g̃ 𝛾 :L → 𝒫(U) × I, x ↦ (f̃ ∩̃ g)(x),
where ̃ ̃ → 𝒫(U), x ↦ f̃ (x) ∩ g(x), f̃ ∩̃ g:L ⋁ 𝜆 ∨ 𝛾:L → I, x ↦ {𝜆(x), 𝛾(x)}.
(2.2)
3. Hybrid subsemigroups and ideals Definition 3.1 Let L be a semigroup. A hybrid structure f̃𝜆 in L over U is called a hybrid subsemigroup of L over U if the following assertions are valid:
(∀x, y ∈ X)
�
f̃ (xy) ⊇ f̃ (x) ∩ f̃ (y), ⋁ 𝜆(xy) ≤ {𝜆(x), 𝜆(y)}
�
(3.1)
.
Example 3.2 Let L = {0, 1, 2, 3, 4, 5} be a semigroup with the following Cayley table: ⋅
0
1
2
3
4
5
0
0
0
0
0
0
0
1
0
1
1
1
1
1
2
0
1
2
3
1
1
3
0
1
1
1
2
3
4
0
1
4
5
1
1
5
0
1
1
1
4
5
Let f̃𝜆 be a hybrid structure in L over U = Z which is given by Table 1.
Table 1. Tabular representation of the hybrid structure f̃𝝀 L
f̃
𝝀
0
Z 2Z
0.2
1 2
8N
0.9
3
4N
0.7
4
8N
0.9
5
4Z
0.6
0.5
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Table 2. Tabular representation of the hybrid structure g̃ 𝜸 L 0
g̃
𝜸
Z
0.2
1
Z
0.2
2
4Z
0.6
3
2Z
0.4
4
4N
0.8
5
4N
0.8
It is easy to verify that f̃𝜆 is a hybrid subsemigroup of L over U = Z. Also the hybrid structure g̃ 𝛾 in L over U = Z which is given by Table 2 is a hybrid subsemigroup of L over U = Z. Definition 3.3 Let L be a semigroup. A hybrid structure f̃𝜆 in L over U is called a hybrid left (resp., right) ideal of L over U if the following assertions are valid:
(∀x, y ∈ X)
(
)
f̃ (xy) ⊇ f̃ (y) (resp., f̃ (xy) ⊇ f̃ (x)) 𝜆(xy) ≤ 𝜆(y) (resp., 𝜆(xy) ≤ 𝜆(x))
(3.2)
.
If a hybrid structure f̃𝜆 in L over U is both a hybrid left ideal and a hybrid right ideal of L over U, we say that f̃𝜆 is a hybrid two-sided ideal of L over U. Example 3.4 Let L = {a, b, c, d} be a semigroup with the following Cayley table: ⋅
a
b
c
d
a
a
a
a
a
b
a
a
a
a
c
a
a
b
a
d
a
a
b
b
Then the hybrid structure f̃𝜆 in L over an initial universe set U = {u1 , u2 , u3 , u4 , u5 } which is given by Table 3 is a hybrid two-sided ideal of L over U.
Table 3. Tabular representation of the hybrid structure f̃𝝀 L
f̃
𝝀
a
{u1 , u2 , u3 , u4 }
0.2
b
{u2 , u3 , u4 }
0.5
c
{u3 }
0.9
d
{u2 , u3 }
0.7
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Obviously, every hybrid left (resp., right) ideal is a hybrid subsemigroup, but the converse is not true in general. In fact, the hybrid subsemigroup f̃𝜆 in Example 3.2 is not a hybrid left ideal of L over U = Z since f̃ (3 ⋅ 5) = f̃ (3) = 4N4Z = f̃ (5) and/or 𝜆(3 ⋅ 5) = 𝜆(3) = 0.7 ≰ 0.6 = 𝜆(5). Also the hybrid subsemigroup g̃ 𝛾 in Example 3.2 is not a hybrid right ideal of L over U = Z since ̃ ̃ and/or 𝛾(3 ⋅ 4) = 𝛾(2) = 0.6 ≰ 0.4 = 𝛾(3). ̃ ⋅ 4) = g(2) = 4N2Z = g(3) g(3 For a nonempty subset A of( L and 𝜀, 𝛿 ∈)𝒫(U) with 𝜀 ⊋ 𝛿, and s, t ∈ [0, 1] with t < s, consider a (𝜀, 𝛿) (𝜀, 𝛿) (t, s) ̃ , 𝜒A(𝜆) ,
hybrid structure 𝜒A(t, s) (f̃𝜆 ) = 𝜒
A(f )
where (𝜀, 𝛿) 𝜒A( :L → 𝒫(U), x ↦ f̃ )
{
if x ∈ A, otherwise,
𝜀 𝛿
and (t, s) 𝜒A(𝜆) :L → I, x ↦
{
t s
if x ∈ A, otherwise,
(𝜀, 𝛿) -characteristic hybrid structure in L over U. (t,s) ) (𝜀, 𝛿) (t, s) is called the -identity hybrid structure in L over 𝜒L((𝜀,f̃ )𝛿) , 𝜒L(𝜆) (t,s)
which is called the
The hybrid structure
(𝜀, 𝛿) ̃ 𝜒L(t, (f ) = s) 𝜆
U. The
(
(𝜀,𝛿) -characteris(t,s)
tic (resp., identity) hybrid structure in L over U with 𝜀 = U, 𝛿 = �, t = 0 and s = 1 is called the charac( ) teristic (resp., identity) hybrid structure in L over U, and is denoted by 𝜒A (f̃𝜆 ): = 𝜒A (f̃ ), 𝜒A (𝜆)
( )) ( resp.𝜒f̃ : = 𝜒f̃ , 𝜒𝜆 . 𝜆
Theorem 3.5 For any nonempty subset A of a semigroup L, the following are equivalent: (i) A is a left (resp., right) ideal of L. (ii) The characteristic hybrid structure 𝜒A (f̃𝜆 ) in L over U is a hybrid left (resp., right) ideal of L over U. Proof Assume that A is a left ideal of L. For any x, y ∈ L, if y ∉ A then 𝜒A (f̃ )(xy) ⊇ � = 𝜒A (f̃ )(y) and 𝜒A (𝜆)(xy) ≤ 1 = 𝜒A (𝜆)(y). If y ∈ A, then xy ∈ A and so 𝜒A (f̃ )(xy) = U = 𝜒A (f̃ )(y) and 𝜒A (𝜆)(xy) = 0 = 𝜒A (𝜆)(y). Therefore 𝜒A (f̃𝜆 ) is a hybrid left ideal of L over U. Similarly, 𝜒A (f̃𝜆 ) is a hybrid right ideal of L over U when A is a right ideal of L. Conversely, suppose that 𝜒A (f̃𝜆 ) is a hybrid left ideal of L over U. Let x ∈ L and y ∈ A. Then 𝜒A (f̃ )(y) = U and 𝜒A (𝜆)(y) = 0, and so 𝜒A (f̃ )(xy) ⊇ 𝜒A (f̃ )(y) = U and 𝜒A (𝜆)(xy) ≤ 0 = 𝜒A (𝜆)(y). Hence xy ∈ A and therefore A is a left ideal of L. Similarly, we can show that if 𝜒A (f̃𝜆 ) is a hybrid right ideal of L over U, ✷ then A is a right ideal of L. Corollary 3.6 For any nonempty subset A of a semigroup L, the following are equivalent: (i) A is a two-sided ideal of L. (ii) The characteristic hybrid structure 𝜒A (f̃𝜆 ) in L over U is a hybrid two-sided ideal of L over U. Theorem 3.7 A hybrid structure f̃𝜆 in L over U is a hybrid subsemigroup of L over U if and only if the nonempty sets L𝜀 : = {x ∈ L ∣ f̃ (x) ⊇ 𝜀} and Lt : = {x ∈ L ∣ 𝜆(x) ≤ t} f̃
𝜆
are subsemigroups of L for all (𝜀, t) ∈ 𝒫(U) × I. Proof Suppose that a hybrid structure f̃𝜆 in L over U is a hybrid subsemigroup of L over U. Assume that L𝜀f̃ ≠ ∅ ≠ Lt𝜆 for all (𝜀, t) ∈ 𝒫(U) × I. Let x, y ∈ L𝜀f̃ ∩ Lt𝜆. Then f̃ (x) ⊇ 𝜀, f̃ (y) ⊇ 𝜀, 𝜆(x) ≤ t and 𝜆(y) ≤ t. It follows from (3.1) that Page 5 of 12
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f̃ (xy) ⊇ f̃ (x) ∩ f̃ (y) ⊇ 𝜀, 𝜆(xy) ≤
⋁
(3.3)
{𝜆(x), 𝜆(y)} ≤ t.
Hence xy ∈ L𝜀f̃ ∩ Lt𝜆, and so L𝜀f̃ and Lt𝜆 are subsemigroups of L. Conversely, assume that the nonempty sets L𝜀f̃ and Lt𝜆 are subsemigroups of L for all (𝜀, t) ∈ 𝒫(U) × I For any x, y ∈ L, let f̃ (x) = 𝜀x and f̃ (y) = 𝜀y. If we put 𝜀: = 𝜀x ∩ 𝜀y, then x, y ∈ L𝜀f̃ and so ⋁ f̃ (xy) ⊇ 𝜀 = 𝜀x ∩ 𝜀y = f̃ (x) ∩ f̃ (y). Now, for any a, b ∈ L, let 𝜆(a) = ta and 𝜆(b) = tb. Taking t: = {ta , tb } ⋁ ⋁ t t implies that a, b ∈ L𝜆. Thus ab ∈ L𝜆, which implies that 𝜆(ab) ≤ t = {ta , tb } = {𝜆(a), 𝜆(b)}. There✷ fore f̃𝜆 is a hybrid subsemigroup of L over U. Note that f̃𝜆 [𝜀, t] = Lf̃ ∩ L𝜆 for all (𝜀, t) ∈ 𝒫(U) × I. Hence we have the following corollary. 𝜀
t
Corollary 3.8 If a hybrid structure f̃𝜆 in L over U is a hybrid subsemigroup of L over U, then the nonempty [𝜀, t]-hybrid cut f̃𝜆 [𝜀, t] is a subsemigroup of L for all (𝜀, t) ∈ 𝒫(U) × I. Theorem 3.9 A hybrid structure f̃𝜆 in L over U is a hybrid left (resp., right) ideal of L over U if and only if the nonempty sets L𝜀f̃ : = {x ∈ L|f̃ (x) ⊇ 𝜀} and Lt𝜆 : = {x ∈ L ∣ 𝜆(x) ≤ t}
are left (resp., right) ideals of L for all (𝜀, t) ∈ 𝒫(U) × I. Proof It is the same as the proof of Theorem 3.7.
✷
For any hybrid structures f̃𝜆 and g̃ 𝛾 in L over U, the hybrid product of f̃𝜆 and g̃ 𝛾 is defined to be a ( ) ̃ 𝜆◦𝛾 in L over U where hybrid structure f̃𝜆 ⊙ g̃ 𝛾 = f̃ ◦̃ g,
̃ = (f̃ ◦̃ g)(x)
� � ⋃ �̃ ̃ f (y) ∩ g(z)
if ∃ y, z ∈ L such that x = yz
x=yz
otherwise
� and
(𝜆◦𝛾)(x) =
� ⋀ ⋁ {𝜆(y), 𝛾(z)}
if ∃ y, z ∈ L such that x = yz
x=yz
1
otherwise
for all x ∈ L. ( ) ( ) ( ) ( ) Proposition 3.10 Let f̃𝜆1 = f̃1 , 𝜆1 , f̃𝜆2 = f̃2 , 𝜆2 , g̃ 1𝛾 = g̃ 1 , 𝛾1 and g̃ 2𝛾 = g̃ 2 , 𝛾2 be hybrid structures in L over U. If f̃𝜆1
