This paper deals with a particular class of hyperstructures called semihypergroups, which is a generalization of semigroups. We introduce the notions of SU-qausi hyperideal, SU-bi-hyperideal, SU-generalized bi-hyperideal, SU-interior hyperideal and discuss several properties of these hyperideals. We also characterize regular and intra-regular semihypergroups by the properties of their SU-hyperideals.
Algebraic hyperstructures are the generalizations of classical algebraic structures. Marty was the pioneer of the hypercompositional structures. In 1999, he introduced the theory of hypergroups in [17], which is a generalized form of group theory. Since then hundreds of papers and several books [3, 6, 27] have been published on hyperstructure theory. In 1970, Koskas proposed the notion of semihypergroup which is generalized form of semigroup. In this paper we will characterize semihypergroups by the properties of their soft union hyperideals.
There are certain problems which are closely related to our real life specially in the fields of economics, social sciences, medical sciences and engineering that possess imprecise data. The traditional mathematical methods cannot cope with these type of problems. In different decades remarkable theories have been introduced for dealing with such type of imprecise data in appritiatable manners. In 1999, Molodtsov [18] gave birth to the theory of soft sets. This is a new mathematical tool for dealing with those uncertainities and imprecise data, which traditional mathematical tools cannot handle. Parametric set in soft sets solved this problem efficientely. Furthermore, soft groups and new operations in soft set theory were studied by Aktas and Cagman [1], Ali et al. [2], Maji, Biswas and Roy [14–16]. In [22], Naz and Shabir worked on fuzzy bipolar soft sets.
After that Sezgin and Atagun [25, 26] introduced some operations on soft sets with a different approach. In [24] Sezgin defined soft intersection-union product and introduced a new ideal to the classical ring theory via soft set theory with the concept of soft union rings, ideals and bi-ideals. In this paper we adopt the concept of soft sets with the approach of Sezgin in [24]. In [20] and [21], Naz and Shabir studied soft semihypergroups and prime soft bi-hyperideals of semihypergroups. Shabir and Tariq in [23] characterized semihypergroups by (εγ, εγ ∨ qσ)- fuzzy hyperideals.
Ideal theory is important not only for the intrinsic interest and purity of its logical structure but because it is a necessary tool in many branches of mathematics and its applications such as in informatics, physics and others. In algebraic structures, ideals have been recently used to design efficient classification systems, see [8–12].
This strong historical background of ideals of different algebraic structures provoke us to judge and investigate the behavior of soft union hyperideals in semihypergroups and observe their different properties in different provided conditions. This paper is organized in the following manners.
In Section 2, we study some basic and introductory concepts of semihypergroup, soft union set, soft union hyperideal. In the following sections, we present SU-quasi-hyperideal, SU-bi-hyperideal, SU-generalized bi-hyperideal, SU-interior hyperideal and discuss several properties of these hyperideals. In last section, we characterize regular and intraregular semihypergroups by the properties of their SU-hyperideals.
Preliminaries
In this section, we review some definitions and results from the theory of hyperstructures and semihypergroups which are basics for the later results. The main results of this section have been taken from [4, 5, 19].
Semihypergroups
For a non-empty set S, a hyperoperation on S is a function ∘ from S × S to P∗ (S); where P∗ (S) is the family of non-empty subsets of S. We shall denote x ∘ y = ∘ (x, y) for every x, y ∈ S.
Definition 1. ([13], Definition 1.1.3) A non-empty set S endowed with hyperoperation ∘ is called hypergroupoid and is denoted by (S, ∘).
Definition 2. ([13], Definition 1.2.1) A hypergroupoid (S, ∘) is called a semihypergroup if for all x, y, z ∈ S
which means that
If x ∈ S and A, B are non-empty subsets of S, then
A non-empty subset A of a semihypergroup S is called subsemihypergroup of S if A ∘ A ⊆ A.
Definition 3. [4] An element e in a semihypergroup S is called identity element if x ∈ e ∘ x = x ∘ e for all x ∈ S.
In [7] Hasankhani defined that a non-empty subset I of a semihypergroup S is called a left (right) hyperideal of S if S ∘ I ⊆ I (I ∘ S ⊆ I, respectively). Then I is called a hyperideal of S, if it is both a left and a right hyperideal of the semihypergroup S.
Definition 4. ([13], Definition 3.1.1) A non-empty subset Q of a semihypergroup S is called a quasi-hyperideal of S if (Q ∘ S) ∩ (S ∘ Q) ⊆ Q.
Definition 5. ([13], Definition 3.1.2) A subsemihypergroup B of a semihypergroup S is called a bi-hyperideal of S if B ∘ S ∘ B ⊆ B.
Definition 6. ([13], Definition 3.2.1) A semihypergroup S is called regular, if for each a ∈ S there exists an element x ∈ S such that a ∈ a ∘ x ∘ a.
Definition 7. ([13], Definition 3.2.2) A semihypergroup S is called intra-regular, if for each a ∈ S there exist x, y ∈ S such that a ∈ x ∘ a ∘ a ∘ y.
Theorem 1.([13], Theorem 3.2.4) The following conditions for a semihypergroup S are equivalent.
S is regular.
For every right hyperideal A and left hyperideal B of S, A ∘ B = A ∩ B.
Theorem 2.([19], Theorem 1.3.5) The following statements are equivalent for a semihypergroup S.
S is intra-regular.
R ∩ L ⊆ (L ∘ R) for every right hyperideal R and every left hyperideal L of S.
Theorem 3.([13], Theorem 3.2.8) The following statements are equivalent for a semihypergroup S.
S is regular and intra-regular.
B ∘ B = B for every bi-hyperideal B of S.
Q ∘ Q = Q for every quasi-hyperideal Q of S.
B1 ∩ B2 = (B1 ∘ B2) ∩ (B2 ∘ B1) for every bi-hyperideals B1 and B2 of S.
R ∩ L ⊆ (R ∘ L) ∩ (L ∘ R) for every right hyperideal R and every left hyperideal L of S.
R (a) ∩ L (a) ⊆ (R (a) ∘ L (a)) ∩ (L (a) ∘ R (a)) for every a ∈ S.
Soft sets
The following concept of soft sets is given by Naz in [19]. Throughout in this paper, U refers to an initial universe, E is a set of parameters, P (U) is the powerset of U and A ⊆ E .
Definition 8. A soft set fA over U is a function defined by
Here fA is called an approximate function. A soft set over U can be represented by the set of ordered pairs
It is clear that a soft set is a parameterized family of subsets of the set U. We shall denote the set of all soft sets over U by S (U).
Definition 9. Let fA, fB ∈ S (U). Then fA is a soft subset of fB denoted by , if fA (x) ⊆ fB (x) for all x ∈ E. Two soft sets fA and fB are called equal soft sets if and . It is denoted by
Definition 10. Let fA, fB ∈ S (U). Then union of fA and fB, denoted by and is defined as , where fA∪B (x) = fA (x) ∪ fB (x) for all x ∈ E.
Definition 11. Let fA, fB ∈ S (U). Then intersection of fA and fB, denoted by and is defined as , where fA∩B (x) = fA (x) ∩ fB (x) for all x ∈ E.
Soft sets of a semihypergroup S over U
Throughout in this paper semihypergroup S plays as a role of set of parameters and A, B, C, D, F are non-empty subsets of S.
Definition 12. ([19], Definition 6.1.1) Let fA and gB be two soft sets of a semihypergroup S over U. Then soft union (SU) product fA ⊙ ∪gB is a soft set of S over U, defined for all x ∈ S,
Let x ∈ S. Define Ax = {(y, z) ∈ S × S : x ∈ y ∘ z}.
For any two soft sets fA and gB of S over U, fA ⊙ ∪gB is defined by
Definition 13. ([19], Definition 6.1.7) Let S be a semihypergroup. If A ⊆ S, then the soft characteristic function is the soft set over U, defined as follows
Proposition 1.([19], Proposition 6.1.9) Let X and Y be non-empty subset of a semihypergroup S. Then the following properties hold:
If Y ⊆ X, then .
and .
Definition 14. ([19], Definition 6.1.10) A soft set fA of a semihypergroup S over U is called a soft union (SU-) subsemihypergroup S over U if for every α ∈ x ∘ y
Proposition 2.([19], Proposition 6.1.11) A soft set fA is an SU-subsemihypergroup of S over U if and only if .
Definition 15. ([19], Definition 6.1.13) Let fA be a soft set of S over U and ξ ∈ P (U). Then lower ξ-inclusion of fA defined as .
Proposition 3.([19], Proposition 6.1.14) Let fA be a soft set of a semihypergroup S over U and ξ ∈ P (U). Then fA is an SU-subsemihypergroup of S over U if and only if each is a subsemihypergroup of S.
Proposition 4.([19], Proposition 6.1.12) A non-empty subset A of a semihypergroup S is a subsemihypergroup of S if and only if the soft characteristic functionis an SU-subsemihypergroup of S over U.
Definition 16. ([19], Definition 6.2.1) Let S be a semihypergroup and A be a non-empty subset of S. Then the soft set fA of S over U is called:
a soft union right (or SU-right) hyperideal of S over U if , for every x, y ∈ S;
a soft union left (or SU-left) hyperideal of S over U if , for every x, y ∈ S;
a soft union hyperideal or SU-hyperideal of S over U (or SU two-sided hyperideal) if it is both an SU-left hyperideal and an SU-right hyperideal.
Proposition 5.([19], Proposition 6.1.3) Let fA, gB, hC, tD, lF ∈ S (U). Then
, generally. However, if S is commutative then
and
and
If tD, lF ∈ S (U) such that and , then .
Lemma 1.([19], Lemma 6.1.8) If is a soft set of a semihypergroup S over U, then .
Proposition 6.([19], Proposition 6.2.6) A soft set fA of a semihypergroup S over U is an SU-left (SU-right) hyperideal of S over U if and only if .
Lemma 2.([19], Lemma 6.2.7) If fA is a soft set of a semihypergroup S over U, then is an SU-left (SU-right) hyperideal of S over U.
Proposition 7.([19], Proposition 6.2.9) If fA is an SU-right hyperideal and gB an SU-left hyperideal of S over U, then .
Proposition 8.([19], Proposition 6.2.3) A non-empty subset A of a semihypergroup S is a hyperideal of S if and only if the soft characteristic function of A is an SU-hyperideal of S over U.
Proposition 9.([19], Proposition 6.2.4) A soft set fA of a semihypergroup S over U is an SU-left (SU-right) hyperideal over U if and only if for each ξ ⊆ P (U), is a left (right) hyperideal of S.
Soft union hyperideals in semihypergroups
In this section we define the notions of soft union (SU-) quasi-hyperideal, soft union (SU-) bi-hyperideal, soft union (SU-) generalized bi-hyperideal and soft union (SU-) interior hyperideal of a semihypergroup S over U. We prove some interesting results by using these notions.
SU-quasi-hyperideals of semihypergroup
Definition 17. A soft set fA of a semihypergroup S over U is called a soft union (SU-) quasi-hyperideal of S over U if
It is clear that every SU-left (SU-right) hyperideal of S is an SU-quasi-hyperideal of S over U. The following example shows that the converse may not be true.
Example 1. Let S ={ a, b, c, d } be a semihypergroup with hyperoperation ∘ defined as
∘
a
b
c
d
a
{a}
{ a }
{ a }
{a}
b
{a}
{a, b}
{a, c}
{a}
c
{a}
{ a }
{a, b}
{a}
d
{a}
{a, d}
{a}
{a}
Let U ={ x, y, z } and A = { c, d } ⊆ S . Define fA (a) = φ, fA (b) = φ, fA (c) ={ x, y }, fA (d) ={ y }. Then fA is an SU-quasi-hyperideal of S over U . But fA is neither SU-left nor SU-right hyperideal of S over U. Because and .
Proposition 10.Every SU-quasi-hyperideal of a semihypergroup S over U is an SU-subsemihypergroup of S over U.
Proof. Let fA be an SU-quasi-hyperideal of S over U. Since so by Proposition 5, and .
Hence Thus by Proposition 2, fA is an SU-subsemihypergroup of S over U. □
Lemma 3.If fA is a soft set of semihypergroup S over U, then is an SU-left (SU-right) hyperideal of S over U.
Proof. Let fA be a soft set of S over U. Then
Thus is an SU-left hyperideal of S over U. Similarly, we can prove that is an SU-right hyperideal of S over U. □
Proposition 11.A non-empty subset Q of a semihypergroup S is a quasi-hyperideal of S if and only if the soft characteristic function of complement of Q is an SU-quasi-hyperideal of S over U.
Proof. Suppose Q is a quasi-hyperideal of a semihypergroup S and is the soft characteristic function of complement of Q over U. Let a be any element of S. If a ∈ Q, then , so we have . If a ∉ Q, then .
Let . Then for every a ∈ x ∘ y, and
This implies and , so there exist elements b, c, d and e of S with a ∈ b ∘ c and a ∈ d ∘ e such that and . Hence a ∈ b ∘ c ⊆ Q ∘ S and a ∈ d ∘ e ⊆ S ∘ Q. This implies that a ∈ (Q ∘ S) ∩ (S ∘ Q) ⊆ Q, which contradicts our supposition that a ∉ Q. Thus we have
. Thus is an SU-quasi-hyperideal of a semihypergroup S over U. Conversely, let be an SU-quasi-hyperideal of S over U. Let a ∈ (Q ∘ S) ∩ (S ∘ Q). Then there exist elements s and t of S and elements b and c of Q such that a ∈ b ∘ s and a ∈ t ∘ c. This implies Aa ≠ φ. Thus we have
Thus . Similarly, we can show that Hence by hypothesis
Which shows that a ∈ Q and so (Q ∘ S) ∩ (S ∘ Q) ⊆ Q. Thus Q is a quasi-hyperideal of S. □
Theorem 4.If a soft set fA of a semihypergroup S over U is an SU-quasi-hyperideal of S over U, then each non-empty lower ξ-inclusion of fA is a quasi-hyperideal of S.
Proof. Suppose fA is an SU-quasi-hyperideal of S over U and ξ ∈ P (U) be such that . Let . Then and . Let and s, t ∈ S be such that a ∈ b ∘ s and a ∈ t ∘ c, so Aa ≠ φ. Thus by hypothesis
This implies and so . Hence is a quasi-hyperideal of S. □
Proposition 12.The union of any family of SU-quasi-hyperideals of a semihypergroup S over U is an SU-quasi-hyperideal of S over U.
Proof. Let {fAi ; i∈ I } be a family of SU-quasi-hyperideal of S over U. Then
for all i ∈ I . Since each fAi is an SU-quasi-hyperideal of S over U, so we have for all i ∈ I . Thus
.
Hence is an SU-quasi-hyperideal of S over U. □
Corollary 1. Let fA and gB be SU-right hyperideal and SU-left hyperideal of a semihypergroup S over U, respectively. Then is an SU-quasi-hyperideal of S over U.
Proof. The proof is straightforward, because every one-sided SU-hyperideal is an SU-quasi-hyperideal of S over U. □
Proposition 13.Every SU-quasi-hyperideal fA of a semihypergroup S over U is the union of an SU-left hyperideal and an SU-right hyperideal of S over U.
Proof. Let fA be an SU-quasi-hyperideal of S over U. By Lemma 3, and are SU-left and SU-right hyperideals of S over U, respectively. Thus by Corollary 3.1, is an SU-quasi-hyperideal of S over U. It is clear that
This shows . □
Soft union bi-hyperideal
In this section, we define soft union bi-hyperideal of a semihypergroup S over U and prove some results.
Definition 18. An SU-subsemihypergroup fA of a semihypergroup S over U is called a soft union(SU-) bi-hyperideal of S over U if for all x, y, z ∈ S .
Proposition 14.A non-empty subset B of a semihypergroup S is a bi-hyperideal of S if and only if the soft characteristic function of complement of B is an SU-bi-hyperideal of S over U.
Proof. Suppose B is a bi-hyperideal of S. Thenby Proposition 4, is an SU-subsemihypergroup of S over U. Let x, y, z ∈ S. If x, z ∈ B, then So for every α ∈ x ∘ y ∘ z ⊆ B ∘ S ∘ B ⊆ B, we have This implies If x ∉ B or z ∉ B, then or . Thus we have . This implies , which shows that is an SU-bi-hyperideal of S over U. Conversely, assume that is an SU-bi-hyperideal of S over U. Then by Proposition 4, B is a subsemihypergroup of S. Let α ∈ B ∘ S ∘ B . Then there exist x, z ∈ B and y ∈ S such that α ∈ x ∘ y ∘ z. Since Hence for each α ∈ x ∘ y ∘ z, we have and so α ∈ B. This implies B ∘ S ∘ B ⊆ B. Therefore B is a bi-hyperideal of S. □
Proposition 15.Let fA be a soft set of a semihypergroup S over U. Then fA is an SU-bi-hyperideal of S over U if and only if
Proof. Suppose fA is an SU-bi-hyperideal of S over U. Then fA is an SU-subsemihypergroup of S over U. Hence by Proposition 2, . Let x ∈ S. If Ax = φ, then If Ax ≠ φ, then
Since (y, z) ∈ Ax, this means x ∈ y ∘ z and (p, q) ∈ Az means that z ∈ p ∘ q . Thus x ∈ y ∘ z ⊆ y ∘ (p ∘ q). Since fA is an SU-bi-hyperideal of S over U, so This implies that fA (x) ⊆ fA (y) ∪ fA (q) for all y, p, q ∈ S. Thus we have . Therefore . Conversely, suppose that and Since , so by Proposition 2, fA is an SU-subsemihypergroup of S over U. Let α ∈ a ∘ x ∘ b, for a, b, x ∈ S,. Then there exists β ∈ a ∘ x such that α ∈ β ∘ b this means that Aα ≠ φ.
Thus, by hypothesis, we have
So we have fA (α) ⊆ fA (a) ∪ fA (b) for every α ∈ a ∘ x ∘ b. Thus. Hence fA is an SU-bi-hyperideal of S over U. □
Theorem 5.A soft set fA of a semihypergroup S over U is an SU-bi-hyperideal of S over U if and only if each non-empty lower ξ-inclusion of fA is a bi-hyperideal of S.
Proof. Suppose fA is an SU-bi-hyperideal of S over U. Then fA is an SU-subsemihypergroup of S over U. By Proposition 3, is a subsemihypergroup of S. Let . Then fA (a) ⊆ ξ and fA (b) ⊆ ξ. Let s ∈ S. Then by hypothesis . This implies fA (α) ⊆ ξ for every α ∈ a ∘ s ∘ b. Thus . This shows that . Hence is a bi-hyperideal of S. Conversely, suppose each non-empty subset of S is a bi-hyperideal of S. Then is a subsemihypergroup of S. By Proposition 3, fA is an SU-subsemihypergroup of S over U. Now we show that . If fA (a) ∪ fA (b) = U, then . If fA (a) ∪ fA (b) ≠ U. Let fA (a) ∪ fA (b) = ξ ⊆ U. Then ξ = fA (a) ∪ fA (b) ⊇ fA (a) and ξ = fA (a) ∪ fA (b) ⊇ fA (b). This implies that . Since each is a bi-hyperideal of S, so for s ∈ S, . Let α ∈ a ∘ s ∘ b. Then fA (α) ⊆ ξ for every α ∈ a ∘ s ∘ b. Thus and hence fA is an SU-bi-hyperideal of S over U. □
Example 2. Let S ={ a, b, c, d } be a semihypergroup with hyperoperation ∘ defined by
∘
a
b
c
d
a
{a}
{a}
{a}
{a}
b
{a}
{a}
{a}
{a}
c
{a}
{a}
{a, b}
{a}
d
{a}
{a}
{a, b}
{a, b}
Let U ={ x, y, z }. Here {a}, {a, b}, {a, b, c}, {a, b, d} and S are bi-hyperideals of S . Let A = { b, c, d } ⊆ S and define fA (a) = φ, fA (b) = { y } , fA (c) = U, fA (d) = { y, z } . Then we have
Since each is a bi-hyperideal of S, so by Theorem 5, fA is an SU-bi-hyperideal of Sover U.
Proposition 16.Every SU-quasi-hyperideal of a semihypergroup S over U is an SU-bi-hyperideal of S over U.
Proof. Let fA be an SU-quasi-hyperideal of S over U. Then by Proposition 10, fA is an SU-subsemihypergroup of S over U. Since and , so we have and . This implies . Since fA is an SU-quasi-hyperideal, so . Thus and hence fA is an SU-bi-hyperideal of S over U. □
Corollary 2.Every one-sided SU-hyperideal of a semihypergroup S over U is an SU-bi-hyperideal of S over U.
But converse of the above Corollary is not true, as shown in the following example.
Example 3. Let S ={ a, b, c, d } be a semihypergroup with hyperoperation ∘ defined by
∘
a
b
c
d
a
{a}
{ a }
{ a }
{a}
b
{a}
{a, b}
{a, c}
{a}
c
{a}
{ a }
{a, b}
{a}
d
{a}
{a, d}
{ a }
{a}
Let U ={ x, y, z }. Here {a } , { a, b } , { a, c } , { a, d } , {a, b, c} , { a, c, d } and S are all possible bi-hyperideals of S. It is clear that {a, b} is neither left nor right hyperideal of S. Let B = { b, c, d } ⊆ S and define fB (a) = φ, fB (b) ={ x, y } and fB (c) = { z } , fB (d) = U. Then we have
Since each is a bi-hyperideal of S, so by Theorem 5, fB is a SU-bi-hyperideal of S over U. But fB is neither SU-left nor SU-right hyperideal of S over U, because {a, b} is neither left nor right hyperideal of S.
Proposition 17.The union a family of SU-bi-hyperideals of a semihypergroup S over U is an SU-bi-hyperideal.
Proof. Let {fAi, i∈ I } be a family of SU-bi-hyperideals of S over U. Then
for each i ∈ I. Since each fAi is an SU-subsemihypergroup of S over U, we have for each i ∈ I. Also for each i ∈ I. Since each fAi is an SU-bi-hyperideal of S over U, we have for each i ∈ I. Thus . Hence is an SU-bi-hyperideal of S over U. □
But the intersection of SU-bi-hyperideal of a semihypergroup S over U is not an SU-bi-hyperideal of S as we have shown in the following example.
Example 4. Let S ={ a, b, c, d } be a semihypergroup under the hyperoperation ∘ defined by
∘
a
b
c
d
a
{a}
{a}
{ a }
{a}
b
{a}
{a}
{a, d}
{a}
c
{a}
{a}
{ a }
{a}
d
{a}
{a}
{ a }
{a}
Here {a}, {a, b},{a, c} {a, d}, {a, b, d}, {a, c, d} and S are bi-hyperideals of S . But {a, b, c} is not a bi-hyperideal. Let U ={ x, y, z } and A ={ b, c, d } be a subset of S. Define the soft set fA by fA (a) = φ, fA (b) ={ x } , fA (c) = U, fA (d) = { y, z }. Now is given by
Since each is a bi-hyperideal of S, so by Theorem 5, fA is an SU-bi-hyperideal of S over U. Define another soft set hA by hA (a) = φ, hA (b) ={ x, y } , hA (c) = { x, z } , hA (d) = { y }. Now is given by
Since each is a bi-hyperideal of S, so by Theorem 5, hA is an SU-bi-hyperideal of S over U.
But is not an SU-bi-hyperideal of S over U, because is defined by , . Then is given by
Since {a, b, c} is not a bi-hyperideal of S, we have is not an SU-bi-hyperideal of S over U.
Proposition 18.The product of two SU-bi-hyperideals of a semihypergroup S over U is an SU-bi-hyperideal of S over U.
Proof. Let fA and gB be SU-bi-hyperideals of a semihypergroup S over U. Then
Now,
This implies fA ⊙ gB is an SU-bi-hyperideal of S over U. □
Soft union generalized Bi-hyperideal
In this section, we introduce soft union generalized bi-hyperideal of a semihypergroup S over U and discuss some results.
Definition 19. A soft set fA of a semihypergroup S over U is called a soft union (SU-) generalized bi-hyperideal of S over U if for all x, y, z ∈ S .
Proposition 19.A non-empty subset G of a semihypergroup S is a generalized bi-hyperideal of S if and only if the SU-characteristic function of complement of G is an SU-generalized bi-hyperideal of S over U.
Proof. The proof is similar to the proof of Proposition 14. □
Proposition 20.Let fG be a soft set of a semihypergroup S over U. Then fG is an SU-generalized bi-hyperideal of S over U if and only if
Proof. The proof is similar to the proof of Proposition 15. □
Theorem 6.A soft set fG of a semihypergroup S over U is an SU-generalized bi-hyperideal of S over U if and only if each non-empty lower ξ-inclusion of fG is a generalized bi-hyperideal of S.
Proof. The proof is similar to the proof of Theorem 5. □
Remark 1. Every SU-bi-hyperideal of a semihypergroup S is an SU-generalized bi-hyperideal of S over U. But the converse is not true, as shown in the following example.
Example 5. Let S ={ a, b, c, d } be a semihypergroup with hyperoperation ∘ defined as
∘
a
b
c
d
a
{a}
{a}
{a}
{a}
b
{a}
{a}
{a}
{a}
c
{a}
{a}
{a, b}
{a}
d
{a}
{a}
{a, b}
{a, b}
Let U ={ x, y, z }. Here {a } , { a, b } , { a, c } , { a, d } ,{a, b, c} , { a, b, d } and S are all generalized bi-hyperideals of S and {a} , { a, b } , { a, b, c } , { a, b, d } and S are bi-hyperideals of S. Let G = { b, c, d } ⊆ S be a generalized bi-hyperideal of S and define soft set fG as fG (a) = φ, fG (b) = { x, y } , fG (c) = { y } , fG (d) = U. Then we have
Since each is a generalized bi-hyperideal of S, so by Theorem 6, fG is an SU-generalized bi-hyperideal of S over U, but not an SU-bi-hyperideal of S over U, because {a, c} is not a bi-hyperideal of S.
Theorem 7.Let fA and gB be SU-generalized bi-hyperideals of S over U. Then fA ⊙ gB is an SU-generalized bi-hyperideal of S over U.
Proof. The proof is similar to the proof of Proposition 18. □
Soft union interior hyperideal
In this section, we define soft union interior hyperideal of a semihypergroup S over U and discuss some of its results.
Definition 20. A soft set fI of a semihypergroup S over U is called a soft union (SU-) interior hyperideal of S over U if for every a, x, y ∈ S.
Proposition 21.A non-empty subset I of a semihypergroup S is an interior hyperideal of S if and only if the soft characteristic function of complement I is an SU-interior hyperideal of S over U.
Proof. The proof is similar to the proof of Proposition 14. □
Proposition 22.Let fI be a soft set of a semihypergroup S over U. Then fI is an SU-interior hyperideal of S over U if and only if .
Proof. The proof is similar to the proof of Proposition 15. □
Proposition 23.Let fI and gJ be two SU-interior hyperideals of a semihypergroup S over U. Then and are SU-interior hyperideals of S over U.
Proof. Since
Thus is an SU-interior hyperideal of S over U. Now . Also . This implies that . Thus is a soft interior hyperideal of S over U. □
Theorem 8.A soft set fI of a semihypergroup S over U is an SU-interior hyperideal of S over U if and only if each non-empty lower ξ-inclusion of fI is an interior hyperideal of S.
Proof. The proof is similar to the proof of Theorem 5. □
Lemma 4.Every SU-hyperideal fI of a semihypergroup S over U is an SU-interior hyperideal of S over U.
Proof. Straightforward. □
The converse of the above Lemma is not true, as shown in the following example.
Example 6. Let S ={ a, b, c, d } be a semihypergroup under the hyperoperation ∘ defined as follows
∘
a
b
c
d
a
{a}
{a}
{ a }
{a}
b
{a}
{a}
{ a, d }
{a}
c
{a}
{a}
{ a }
{a}
d
{a}
{a}
{ a }
{a}
Then all possible interior hyperideals of S are {a } , {a, b} , { a, c } , { a, d } , { a, b, d } , { a, c, d } and S but {a, b} is not a hyperideal of S. Let U ={ x, y, z } be the initial universe and I = { b, c, d } ⊆ S. Define soft set fI as fI (a) = φ, fI (b) ={ x } and fI (c) ={ x, y, z } , fI (d) = { x, y }. Then we have
Since each is an interior hyperideal of S, so by Theorem 8, fI is an SU-interior hyperideal of S over U, but fI is not an SU-hyperideal because {a, b} is not a hyperideal of S.
Lemma 5.Let S be a semihypergroup with identity e. Then fI is an SU-hyperideal of S over U if and only if fI is an SU-interior hyperideal of S over U.
Proof. Let fI be an SU-hyperideal of S over U. By Lemma 4, fI is an SU-interior hyperideal of S over U. Now, let fI be an SU-interior hyperideal of S over U and x, y ∈ S. Let α ∈ x ∘ y. Since e is the identity element in S, so α ∈ e ∘ α ⊆ e ∘ x ∘ y. Thus . Similarly, . Hence fI is an SU-hyperideal of S over U. □
Regular and intra-regular semihypergroups
In this section, we characterize regular and intra-regular semihypergroups by the properties of their soft union hyperideals.
Theorem 9.A semihypergroup S is regular if and only if for every SU-right hyperideal fA and SU-left hyperideal gB of S over U .
Proof. Suppose S is regular semihypergroup. Let fA and gB be SU-right and SU-left hyperideals of S over U, respectively. Let a ∈ S. Then there exists x ∈ S such that a ∈ a ∘ x ∘ a. Let α ∈ x ∘ a be such that a ∈ a ∘ α. Then (a, α) ∈ Aa, this implies Aa ≠ φ. Thus we have
This implies and by Proposition 7, . So, . Conversely, assume that for every SU-right hyperideal fA and SU-left hyperideal gB of S over U . Let A and B be right and left hyperideals of S, respectively. Then by Proposition 8, the soft characteristic functions and of complement of A and B are SU-right hyperideal and SU-left hyperideals of S over U, respectively. Hence by hypothesis using Proposition 1, we have . This implies (A ∘ B) c = Ac ∪ Bc = (A ∩ B) c. Thus A ∘ B = A ∩ B. Hence by Proposition 1, S is a regular semihypergroup. □
Theorem 10.The following assertions are equivalent for a semihypergroup S.
S is regular.
for every SU-generalized bi-hyperideal fA of S over U.
for every SU-bi-hyperideal fA of S over U.
for every SU-quasi-hyperideal fA of S over U.
Proof. (i) ⇒ (ii) Let S be a regular semihypergroup and fA be an SU-generalized bi-hyperideal of S over U . For a ∈ S, there exists s ∈ S such that a ∈ a ∘ s ∘ a. Let α ∈ a ∘ s be such that a ∈ α ∘ a, so Aa ≠ φ. Then
Thus we have . Since fA is an SU-generalized bi-hyperideal of S over U, so . Hence . (ii) ⇒ (iii) and (iii) ⇒ (iv) are straightforward. (iv) ⇒ (i) Let fA and gB be SU-right and SU-left hyperideals of S over U, respectively. Then by Corollary 1, is an SU-quasi-hyperideal of S over U. Hence by hypothesis But always true. Hence . Thus by Theorem 9, S is a regular semihypergroup. □
Theorem 11.The following assertions are equivalent for a semihypergroup S .
S is regular.
⊙fA for every SU-quasi-hyperideal fA and every SU-two-sided hyperideal gB of S over U.
for every SU-quasi-hyperideal fA and every SU-interior hyperideal gB of S over U.
for every SU-bi-hyperideal fA and every SU-two-sided hyperideal gB of S over U.
for every SU-bi-hyperideal fA and every SU-interior hyperideal gB of S over U.
for every SU-generalized bi-hyperideal fA and every SU-two-sided hyperideal gB of S over U.
for every SU-generalized bi-hyperideal fA and every SU-interior hyperideal gB of S over U.
Proof. (i) ⇒ (vii) Let fA and gB be any SU-generalized bi-hyperideal and SU-interior hyperideal of S over U, respectively. Then and . Thus Let a ∈ S. Then there exists x ∈ S such that a ∈ a ∘ x ∘ a ⊆ a ∘ (x ∘ a ∘ x ∘ a) = a ∘ ((x ∘ a ∘ x) ∘ a) . Let α ∈ (x ∘ a ∘ x) ∘ a, β ∈ x ∘ a ∘ x. Then α ∈ β ∘ a and a ∈ a ∘ α, that is, (a, α) ∈ Aa. Thus Aa ≠ φ, so we have
Also , so we have (fA ⊙ gB ⊙ fA) (a) ⊆ fA (a) ∪ gB (a) . This implies . Hence . It is clear that
(vii) ⇒ (v) ⇒ (iii) ⇒ (ii) and (vii) ⇒ (vi) ⇒ (iv) ⇒ (ii) . (ii) ⇒ (i) Let fA be an SU-quasi-hyperideal of S over U and is an SU-two-sided hyperideal of S over U. Then . Therefore by Theorem 10, S is regular. □
Theorem 12.The following assertions are equivalent for a semihypergroup S .
S is regular.
for every SU-generalized bi-hyperideal fA and every SU-left hyperideal gB of S over U.
for every SU-bi-hyperideal fA and every SU-left hyperideal gB of S over U.
for every SU-quasi-hyperideal fA and every SU-left hyperideal gB of S over U.
Proof. (i) ⇒ (ii) Let fA and gB be any SU-generalized bi-hyperideal and SU-left hyperideal of S over U, respectively. Then for a ∈ S there exists x ∈ S such that a ∈ a ∘ x ∘ a. Let α ∈ x ∘ a be such that a ∈ a ∘ α, so (a, α) ∈ Aa, that is Aa ≠ φ. Thus ⊆fA (a) ∪ gB (α) . Since gB is an SU-left hyperideal of S over U, so . Thus we have Hence . (ii) ⇒ (iii) implies (iv) are straightforward. (iv) ⇒ (i) Let fA and gB be any SU-right hyperideal and SU-left hyperideal of S over U, respectively. Then fA is an SU-quasi-hyperideal of S over U. Thus by hypothesis . But always holds. Thus . Therefore by Theorem 9, S is a regular semihypergroup. □
Theorem 13.The following assertions are equivalent for a semihypergroup S .
S is regular.
for every SU-right hyperideal fA and every SU-generalized bi-hyperideal gB of S over U.
for every SU-right hyperideal fA and every SU-bi-hyperideal gB of S over U.
for every SU-right hyperideal fA and every SU-quasi-hyperideal gB of S over U.
Proof. The proof is similar to the proof of Theorem 12. □
Theorem 14.The following assertions are equivalent for a semihypergroup S .
S is regular.
for every SU-right hyperideal fA, every SU-generalized bi-hyperideal gB and for every SU-left hyperideal hC of S over U.
for every SU-right hyperideal fA, every SU-bi-hyperideal gB and for every SU-left hyperideal hC of S over U.
for every SU-right hyperideal fA, every SU-quasi-hyperideal gB and for every SU-left hyperideal hC of S over U.
Proof. (i) ⇒ (ii) Let fA, gB and hC be any SU-right hyperideal, SU-generalized bi-hyperideal and SU-left hyperideal of S over U, respectively. For a ∈ S there exists x ∈ S such that a ∈ a ∘ x ∘ a. Let α ∈ a ∘ x be such that a ∈ α ∘ a, that is, (α, a) ∈ Aa. Thus Aa ≠ φ, so we have
Since a ∈ a ∘ x ∘ a. Let β ∈ x ∘ a be such thata ∈ a ∘ β, that is, (a, β) ∈ Aa. Thus Aa ≠ φ, so we have . Since fA is an SU-right hyperideal of S over U, so Also hC is an SU-left hyperideal of S over U, so . Thus we have (fA ⊙ gB ⊙ hC) (a) ⊆ fA (a) ∪ gB (a) ∪ hC (a) . Hence . (ii) ⇒ (iii) implies (iv) are straightforward. (iv) ⇒ (i) Let fA and gB be any SU-right hyperideal and SU-left hyperideal of S over U, respectively. Since is an SU-quasi-hyperideal of S over U, so we have This implies . But is always true, for every SU-right hyperideal fA and SU-left hyperideal gB of S over U. Therefore . Hence by Theorem 9, S is a regular semihypergroup. □
Theorem 15.Let S be a semihypergroup. Then the following conditions are equivalent.
S is intra-regular.
L ∩ R ⊆ L ∘ R for every right hyperideal R and every left hyperideal L of S.
for every SU-right hyperideal gB and SU-left hyperideal fA of S over U.
Proof. (i) ⇒ (iii) Let fA and gB be any SU-left hyperideal and any SU-right hyperideal of S over U, respectively. Let a ∈ S. Since S is intra-regular, so there exist elements x, y ∈ S such that a ∈ x ∘ a ∘ a ∘ y. Let α ∈ x ∘ a and β ∈ a ∘ y be such that a ∈ α ∘ β. Then Aa ≠ φ, so we have ⊆ fA (α) ∪ gB (β). Since fA and gB are SU-left hyperideal and SU-right hyperideal of S over U, respectively, we have and . Thus , that is . (iii) ⇒ (ii) Let L and R be any left and right hyperideals of S, respectively. Then by Proposition 8, and are SU-left and SU-right hyperideals of S over U, respectively. By hypothesis, we have and by Proposition 1, we have . This implies L ∩ R ⊆ L ∘ R. (ii) ⇔ (i) This part is proved in Theorem 2. □
Now we characterize regular and intra regular semihypergroups by the properties of their soft union hyperideals.
Theorem 16.Let S be a semihypergroup. Then the following statements are equivalent:
S is regular and intra-regular.
Every SU-quasi-hyperideal of S over U is idempotent.
Every SU-bi-hyperideal of S over U is idempotent.
for every SU-quasi-hyperideal fA and gB of S over U.
for every SU-quasi-hyperideal fA and every SU-bi-hyperideal gB of S over U.
for every SU-quasi-hyperideal fA and every SU-generalized bi-hyperideal gB of S over U.
for every SU-bi-hyperideal fA and every SU-quasi-hyperideal gB of S over U.
for every SU-bi-hyperideal fA and gB of S over U.
for every SU-bi-hyperideal fA and every SU-generalized bi-hyperideal gB of S over U.
for every SU-generalized bi-hyperideal fA and every SU-quasi-hyperideal gB of S over U.
for every SU-generalized bi-hyperideal fA and every SU-bi-hyperideal gB of S over U.
for every SU-generalized bi-hyperideal fA and gB of S over U.
Proof. (i) ⇒ (xii) Let fA and gB be any SU-generalized bi-hyperideals of S over U and a ∈ S. Then by hypothesis, there exist x, y, z ∈ S such that a ∈ a ∘ x ∘ a and a ∈ y ∘ a ∘ a ∘ z. Thus a ∈ a ∘ x ∘ a ⊆ a ∘ x ∘ (a ∘ x ∘ a) ⊆ (a ∘ x) ∘ a ∘ (x ∘ a)⊆ (a ∘ x) ∘ (y ∘ a ∘ a ∘ z) ∘ (x ∘ a) = (a ∘ x ∘ y ∘ a) ∘ (a ∘ z ∘ x ∘ a). Then for some p ∈ a ∘ x ∘ y ∘ a and q ∈ a ∘ z ∘ x ∘ a, we have a ∈ p ∘ q, that is, (p, q) ∈ Aa. Since Aa ≠ φ, so we have ⊆ fA (p) ∪ gB (q). As fA is an SU-generalized bi-hyperideal of S over U, so and Thus we have Hence . It is clear that (xii) ⇒ (xi) ⇒ (x) ⇒ (iv) ⇒ (ii), (xii) ⇒ (ix) ⇒ (viii) ⇒ (vii) ⇒ (iv), (xii) ⇒ (vi) ⇒ (v) ⇒ (iv) and (viii) ⇒ (iii) ⇒ (ii). (ii) ⇒ (i) Let Q be a quasi-hyperideal of S. Since by Proposition 11, the soft characteristic function of complement of Q is an SU-quasi-hyperideal of S. From Proposition 1, we have This implies (Q ∘ Q) c = Qc, that is Q = Q ∘ Q. Hence by Theorem 3, S is regular and intra-regular. □
Theorem 17.Let S be a semihypergroup. Then the following statements are equivalent:
S is both regular and intra-regular.
for every SU-right hyperideal fA and every SU-left hyperideal gB of S over U .
for every SU-right hyperideal fA and every SU-quasi-hyperideal gB of S over U .
for every SU-right hyperideal fA and every SU-bi-hyperideal gB of S over U .
for every SU-right hyperideal fA and every SU-generalized bi-hyperideal gB of S over U .
for every SU-left hyperideal fA and every SU-quasi-hyperideal gB of S over U .
for every SU-left hyperideal fA and every SU-bi-hyperideal gB of S over U .
for every SU-left hyperideal fA and every SU-generalized bi-hyperideal gB of S over U .
for every SU-quasi-hyperideals fA and gB of S over U .
for every SU-quasi-hyperideal fA and every SU-bi-hyperideal gB of S over U .
for every SU-quasi-hyperideal fA and every SU-generalized bi-hyperideal gB of S over U .
for every SU-bi-hyperideals fA and gB of S over U .
for everySU-bi-hyperideal fA and every SU-generalized bi-hyperideal gB of S over U .
for every SU-generalized bi-hyperideals fA and gB of S over U .
Proof. (i) ⇒ (xiv) Let fA and gB be SU-generalized bi-hyperideals of S over U . Then by Theorem 16, . Similarly Therefore we have . It is clear that(xiv) ⇒ (xiii) ⇒ (xii) ⇒ (ix) ⇒ (vi) ⇒ (ii), (xiv) ⇒ (xi) ⇒ (x) ⇒ (ix) , (xiv) ⇒ (v) ⇒ (iv) ⇒ (iii) ⇒ (ii) , (xiv) ⇒ (viii) ⇒ (vii) ⇒ (vi) ⇒ (ii). (ii) ⇒ (i) Let fA and gB be any SU-right hyperideal and SU-left hyperideal of S over U, respectively. Then we have . Thus by Theorem 15, S is intra-regular. Since . Also , always holds for every SU-right hyperideal and SU-left hyperideal. Therefore we have Thus by Theorem 9, S is regular. □
Theorem 18.Let S be a semihypergroup. Then the following statements are equivalent:
S is regular and intra-regular.
for every SU-quasi-hyperideal fA and every SU-left hyperideal gB of S over U.
for every SU-quasi-hyperideal fA and every SU-right hyperideal gB of S over U.
for every SU-quasi-hyperideals fA and gB of S over U.
for every SU-quasi-hyperideal fA and every SU-bi-hyperideal gB of S over U.
for every SU-quasi-hyperideal fA and every SU-generalized bi-hyperideal gB of S over U.
for every SU-bi-hyperideal fA and every SU-left hyperideal gB of S over U.
for every SU-bi-hyperideal fA and every SU-right hyperideal gB of S over U.
for every SU-bi-hyperideal fA and every SU-quasi-hyperideal gB of S over U.
for every SU-bi-hyperideals fA and gB of S over U.
for every SU-bi-hyperideal fA and every SU-generalized bi-hyperideal gB of S over U.
for every SU-generalized bi-hyperideal fA and every SU-left hyperideal gB of S over U.
for every SU-generalized bi-hyperideal fA and every SU-right hyperideal gB of S over U.
for every SU-generalized bi-hyperideal fA and every SU-quasi-hyperideal gB of S over U.
for every SU-generalized bi-hyperideal fA and every SU-bi-hyperideal gB of S over U.
for every SU-generalized bi-hyperideals fA and gB of S over U.
Proof. (i) ⇒ (xvi) Let fA and gB be any SU-generalized bi-hyperideals of S over U and a ∈ S. Since S is regular as well as intra-regular semihypergroup, so there exist x, y, z ∈ S such that a ∈ a ∘ x ∘ a and a ∈ y ∘ a ∘ a ∘ z. Thus a ∈ a ∘ x ∘ a ⊆ (a ∘ x ∘ a) ∘ x ∘ (a ∘ x ∘ a) ⊆a ∘ x ∘ (y ∘ a ∘ a ∘ z) ∘ x ∘ (y ∘ a ∘ a ∘ z) ∘ x ∘ a = (a ∘ x ∘ y ∘ a) ∘ (a ∘ z ∘ x ∘ y ∘ a) ∘ (a ∘ z ∘ x ∘ a). Then for some p ∈ a ∘ x ∘ y ∘ a and q ∈ a ∘ z ∘ x ∘ y ∘ a, r ∈ a ∘ z ∘ x ∘ a, we have a ∈ p ∘ q ∘ r. Let α ∈ p ∘ q be such that a ∈ α ∘ r, that is, (α, r) ∈ Aa. Then Aa ≠ φ, thus we have ⊆ (fA ⊙ gB) (α) ∪ fA (r) ⊆ fA (p) ∪ gB (q) ∪ fA (r). Since fA and gB are SU-generalized bi-hyperideals of S over U, so we have and This shows (fA ⊙ gB ⊙ fA) (a) ⊆fA (r) ∪ gB (p) ∪ fA (q) ⊆ fA (a) ∪ gB (a) ∪ fA (a) = fA (a) ∪ gB (a) Hence. It is clear that (xi) ⇒ (vi) ⇒ (v) ⇒ (iv) ⇒ (iii), (xvi) ⇒ (xv) ⇒ (xiv) ⇒ (xiii) ⇒ (viii) ⇒ (iii), (xiv) ⇒ (xii) ⇒ (vii) ⇒ (ii) and (xvi) ⇒ (xi) ⇒ (x) ⇒ (ix) ⇒ (viii) . (iii) ⇒ (i) Let fA be an SU-quasi-hyperideal of S over U. Since S is an SU-right hyperideal of S over U, we have Thus by Theorem 10, S is regular. Let fA and gB be any SU-left hyperideal and any SU-right hyperideal of S over U, respectively. Since every SU-left hyperideal is an SU-quasi-hyperideal of S over U, so fA is an SU-quasi-hyperideal of S over U. Thus we have by hypothesis. Hence by Theorem 18, S is intra-regular. (ii) ⇒ (i) For S is regular, proof is similar as in (iii) ⇒ (i). Now let fA be an SU-left hyperideal of S over U and gB be an SU-right hyperideal of S over U. Since every SU-right hyperideal is an SU-quasi-hyperideal of S over U, so gB is an SU-quasi-hyperideal of S over U. Then we have by hypothesis . Hence by Theorem 18, S is intra-regular. □
Theorem 19.Let S be a semihypergroup. Then the following conditions are equivalent:
S is regular and intra-regular.
for every SU-generalized bi-hyperideal fA, every SU-right hyperideal gB and every SU-left hyperideal hC of S over U.
for every SU-bi-hyperideal fA, every SU-right hyperideal gB and every SU-left hyperideal hC of S over U.
for every SU-quasi-hyperideal fA, every SU-right hyperideal gB and every SU-left hyperideal hC of S over U.
Proof. The proof is similar to the proof of Theorem 18. □
Conclusion
In this paper, we have characterized regular and intra-regular semihypergroups in terms of verious types of soft union hyperideals. We can also work on the characterized of different kinds of semihypergroups with the properties of left pure hyperideals and soft left pure hyperideals.
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