A Study of Ordered Ag-Groupoids in terms of Semilattices via Smallest (Fuzzy) Ideals

Abstract

An ordered AG-groupoid can be referred to as an ordered left almost semigroup, as the main difference between an ordered semigroup and an ordered AG-groupoid is the switching of an associative law. In this paper, we define the smallest one-sided ideals in an ordered AG-groupoid and use them to characterize a strongly regular class of a unitary ordered AG-groupoid along with its semilattices and fuzzy one-sided ideals. We also introduce the concept of an ordered -groupoid and investigate its structural properties by using the generated ideals and fuzzy one-sided ideals. These concepts will verify the existing characterizations and will help in achieving more generalized results in future works.

1. Introduction

The concept of fuzzy set was first proposed by Zadeh [1] in . Fuzzy algebra is becoming popular day by day due to wide applications of fuzzification in almost every field. It gives us a tool to model the uncertainty present in phenomena that do not have sharp boundaries. Many papers on fuzzy sets have been published which show its importance and applications to set theory, algebra, real analysis, measure theory, topology, etc. Several algebraists extended the concepts and results of algebra to the broader frame work of fuzzy set theory. Rosenfeld [2] was the first to consider the case when is a groupoid. Kuroki and Mordeson have widely explored fuzzy semigroups in [3, 4]. F. Yousafzai et. al. have also studied the structural properties of certain classes of an ordered LA-semigroup and characterized them in terms of different fuzzy ideals as well [5–8].

In the present paper, we introduce and investigate the notions of smallest left (right) ideals and study the relationship between these ideals along with fuzzy left (right) ideals. As an application of our results we get characterizations of a strongly regular class of a unitary ordered AG-groupoid (an ordered -groupoid) in terms of its semilattices, one-sided (two-sided) ideals based on fuzzy sets, and generated commutative monoids.

2. Preliminaries

An Abelian-Grassmann-groupoid (AG-groupoid) is a nonassociative and a noncommutative algebraic structure lying in a grey area between a groupoid and a commutative semigroup. Commutative law is given by in ternary operations. By putting brackets on the left of this equation, i.e., , in , M. A. Kazim and M. Naseeruddin introduced a new algebraic structure called a left almost semigroup abbreviated as an LA-semigroup [9]. This identity is called the left invertive law. P. V. Protic and N. Stevanovic called the same structure an Abel-Grassmann’s groupoid abbreviated as an AG-groupoid [10].

This structure is closely related to a commutative semigroup because a commutative AG-groupoid is a semigroup [11]. It was proved in [9] that an AG-groupoid is medial; that is, holds for all . An AG-groupoid may or may not contain a left identity. The left identity of an AG-groupoid permits the inverses of elements in the structure. If an AG-groupoid contains a left identity, then this left identity is unique [11]. In an AG-groupoid with left identity (unitary AG-groupoid), the paramedial law holds for all . By using medial law with left identity, we get for all . An AG-groupoid satisfying the identities and for all is called an -groupoid (cf. [9, 12, 13]). We should genuinely acknowledge that much of the ground work has been done by M. A. Kazim, M. Naseeruddin, Q. Mushtaq, M.S. Kamran, P.V. Protic, N. Stevanovic, M. Khan, W.A. Dudek, and R.S. Gigon. One can be referred to [10, 11, 14–19] in this regard.

An AG-groupoid together with a partial order on that is compatible with an AG-groupoid operation, meaning that for , and , is called an ordered AG-groupoid [7].

Let us define a binary operation “” -sandwich operation on an ordered AG-groupoid with left identity as follows:

Then , becomes an ordered semigroup [7].

Note that an ordered AG-groupoid is the generalization of an ordered semigroup because if an ordered AG-groupoid has a right identity then it becomes an ordered semigroup.

Let ; we denote by for some If , then we write For , we denote

(1) A nonempty subset of is called an AG-subgroupoid of if .

(2) A nonempty subset of an ordered AG-groupoid is called a left right ideal of if we have the following

(i) ;

(ii) if and such that , then

Equivalently, a nonempty subset of an ordered AG-groupoid is called left right ideal of if

(3) By two-sided ideal or, simply, ideal, we mean a nonempty subset of an ordered AG-groupoid which is both left and right ideal of .

Lemma 1 (see [7]). Let be an ordered -groupoid and Then the following hold(i);(ii)If , then ;(iii);(iv);(v);(vi), for every ideal of ;(vii), if has a left identity.

3. Fuzzy Set Theory

A fuzzy subset or a fuzzy set of a nonempty set is an arbitrary mapping , where is the unit segment of real line. A fuzzy subset is a class of objects with grades of membership having the form , .

The product of any fuzzy subsets and of is defined by

where

The order relation between any two fuzzy subsets and of is defined by if and only if ,

The symbols and will mean the following fuzzy subsets of : and

For , we define . For , we usually write (1)A fuzzy set of an ordered AG-groupoid is called(i)a fuzzy left ideal of if it satisfies the following(a);(b) ;(ii)a fuzzy right ideal of if it satisfies(a);(b) , (iii)a fuzzy ideal of , if it is both fuzzy left ideal and fuzzy right ideal of .(2)Let be a nonempty subset of We denote by the characteristic function of and define it as follows: (3)A fuzzy subset of is called -semiprime if , .

Lemma 2. Let be any right (left) ideal of an ordered -groupoid . Then is semiprime if and only if is -semiprime.

Proof. Let be a right (left) ideal of ; then, by Lemma 3, is a fuzzy left ideal of . Let , and then , and therefore , and this implies Thus and therefore is semiprime. Converse is simple.

The following results are available in [7, 8].

Lemma 3. For a nonempty subset of an ordered -groupoid , the following conditions are equivalent:(i) is a left (right) ideal of ;(ii)the characteristic function of is a fuzzy left (right) ideal of

Lemma 4. Let be any fuzzy subset of an ordered -groupoid . Then is a fuzzy left (right) ideal of if and only if

Remark 5. The set forms an ordered -groupoid and satisfies all the basic laws.

Remark 6. If is an ordered -groupoid, then

Lemma 7. Let be an ordered -groupoid. For , the following assertions hold:(i)(ii)(iii)(iv)

4. On Strongly Regular Ordered AG-Groupoids

By a unitary ordered AG-groupoid, we shall mean an ordered AG-groupoid with left identity unless otherwise specified.

4.1. Basic Results

This section contains some basic results which will be essential for upcoming section.

Lemma 8. Let be a right ideal and be a left ideal of a unitary ordered -groupoid . Then is a left ideal of

Proof. Let be a right ideal and be a left ideal of Then, by using Lemma 1, we get , which shows that is a left ideal of

The converse inclusion of above lemma is not true in general. Let us consider a unitary ordered AG-groupoid with the following multiplication table and order below.

Set and Then it is easy to verify that is a left ideal of but resp., is not a right (resp., left) ideal of

Lemma 9. Let be a unitary ordered -groupoid. Then is the smallest right ideal of containing , for all .

Proof. Let . Then, by using Lemma 1, we have which shows that is a right ideal of It is easy to see that Let be another right ideal of containing Since hence is the smallest right ideal of containing

Lemma 10. Let be a unitary ordered -groupoid and for all Then becomes a commutative monoid.

Proof. It is simple.

Corollary 11. is the smallest right ideal of an ordered commutative monoid containing , for all .

Lemma 12. Let be a unitary ordered -groupoid and Then is the smallest left ideal of containing

Proof. It is simple.

Theorem 13. Let be a unitary ordered -groupoid and . Then the following assertions hold(i) forms a semilattice.(ii) is a singleton set, if

Proof. It is simple.
Let Then, by using , we get

(i) Recall that an ordered -groupoid is an ordered AG-groupoid in which Note that an ordered -groupoid also satisfies the paramedial law as well.

Now let us introduce the concept of an ordered -groupoid as follows:

(ii) An ordered -groupoid is called an ordered -groupoid if

Note that every unitary ordered AG-groupoid is an ordered -groupoid but the converse is not true in general.

Corollary 14. Let be an ordered -groupoid and . Then the following assertions hold(i) forms a semilattice, where ;(ii) is a singleton set if ,

Lemma 15. Let be an ordered -groupoid and . Then is the right ( left) ideal of

Proof. Letting , then by using Lemma 1, we get which is what we set out to prove. Similarly we can prove that

(i) An element of an ordered AG-groupoid is called a strongly regular element of , if there exists some in such that and , where is called a pseudoinverse of a. is called strongly regular ordered AG-groupoid if all elements of are strongly regular.

Theorem 16. Let be a unitary ordered -groupoid (an ordered -groupoid). An element of is strongly regular if and only if for some .

Proof. (Necessity). Let be strongly regular; then , where Thus for some .
(Sufficiency). Let such that for some ; then , where Thus , and Thus is strongly regular.

Remark 17 (see [7]). Every fuzzy right ideal of a unitary ordered -groupoid is a fuzzy left ideal of but the converse inclusion is not true in general.

Lemma 18. Let be any fuzzy left (right) ideal of a strongly regular unitary ordered -groupoid . Then the following assertions hold:(i), (ii) is -semiprime.

Proof. It is simple.

4.2. Characterization Problems

In this section, we generalize the results of an ordered semigroup and get some interesting characterizations which we usually do not find in an ordered semigroup.

From now onward, will denote any right ( left) ideal of an ordered AG-groupoid ; will denote any smallest right ( smallest left) ideal of containing Any fuzzy left ideal (, fuzzy right ideal) of an ordered AG-groupoid will be denoted by unless otherwise specified.

Theorem 19. Let be any fuzzy left ideals of a unitary ordered -groupoid . Then the following conditions are equivalent:(i) is strongly regular.(ii), (iii)(iv)(v) is strongly regular and , , (vi) is strongly regular and is semilattice.

Proof. It can follow from Theorem 13.
It can follow from Theorem 13.
Let and be any fuzzy left ideals of a strongly regular . Now for , there exist some such that Thus . Therefore which shows that . By using Lemmas 4 and 18, it is easy to show that Thus .
Let and be any right and left ideals of , respectively. Then, by using Lemmas 3 and 8, we have the following.
and are the fuzzy left ideals of . Now by using Lemma 7, we get , which give us Now, by using Lemma 1, we get It is simple.
Since is the smallest right ideal of containing and is the smallest left ideal of containing , , thus, by using given assumption and Lemma 1, we get Hence, by using Theorem 16, is strongly regular.

Theorem 20. Let be a unitary ordered -groupoid. Then the following conditions are equivalent:(i) is strongly regular.(ii), (iii)(iv)(v) is strongly regular and , , (vi) is strongly regular and is semilattice.

Proof. It can follow from Theorem 13.
It can follow from Theorem 13.
Let and be any fuzzy right ideal and fuzzy left ideal of a strongly regular , respectively. From Lemma 4, it is easy to show that . Now, for , there exist some such that Thus Therefore which shows that . Thus .
: Let and be any right and left ideals of , respectively. Then, by using Lemma 3, and are the fuzzy right ideal and fuzzy left ideal of , respectively. Now, by using Lemma 7, we get which implies that . Now by using Lemma 1, we get .
It is simple.
is the smallest right ideal of containing and is the smallest left ideal of containing Thus, by using given assumption and Lemma 1, we get Hence, by using Theorem 16, is strongly regular.

Let be an ordered -groupoid. From now onward, will denote any right left ideal of ; will denote a right ideal of containing and will denote a left ideal of containing ; will denote any fuzzy left ideal (, fuzzy right ideal) of unless otherwise specified.

Theorem 21. Let be an ordered -groupoid. Then is strongly regular if and only if and is semiprime.

Proof. (Necessity). Let be strongly regular. It is easy to see that Let Then there exist some such that which shows that It is easy to see that is semiprime.
(Sufficiency). Since and are the right and left ideals of containing and , respectively, thus, by using given assumption and Lemma 1, we get This implies that is strongly regular.

Corollary 22. Let be an ordered -groupoid. Then is strongly regular if and only if and is semiprime.

Theorem 23. Let be an ordered -groupoid. Then the following conditions are equivalent:(i) is strongly regular.(ii) and is semiprime.(iii) and semiprime.(iv) and is -semiprime.(v) is strongly regular and , , .(vi) is strongly regular and is semilattice.

Proof. It can follow from Corollary 14.
It can follow from Corollary 14.
Let and be any fuzzy right ideal and fuzzy left ideal of a strongly regular , respectively. From Lemma 4, it is easy to show that Now, for , there exist some such that Thus . Therefore which shows that Hence . Also, by using Lemma 18, is -semiprime.
Let and be any left and right ideals of Then, by using Lemma 3, and are the fuzzy right ideal and fuzzy left ideal of . Now, by using Lemma 7, we get , which implies that .
It is simple.
It can follow from Corollary 22.

Theorem 24. Let be an ordered -groupoid. Then the following conditions are equivalent:(i) is strongly regular.(ii) and is semiprime.(iii) and is semiprime.(iv)and is -semiprime.(v) is strongly regular and , (vi) is strongly regular and is semilattice.

Proof. It can follow from Corollary 14.
It can follow from Corollary 14.
Let and be any fuzzy left ideals of a strongly regular . Now, for , there exist some such that
Thus . Therefore which shows that . By using Lemmas 4 and 18, it is easy to show that Thus . Also, by using Lemma 18, is -semiprime.
Let and be any left and right ideals of Then, by Lemma 3, and are the fuzzy right ideal and fuzzy left ideal of , respectively. Now by using Lemmas 7, 8, and 1, we get which shows that . Also, by using Lemma 2, is semiprime.
It is simple.
Since and are the right and left ideals of containing and , respectively, thus by using given assumption and Lemma 1, we get Hence is strongly regular.

5. Conclusions

We have got some interesting and new characterizations which we usually do not find in other algebraic structures. This paper is the continuation of the work carried out in [7, 8]. We introduce the concept of an ordered -groupoid and characterize its strongly regular class in terms of fuzzy left/right ideals via its semilattices and generated commutative monoids.

Some important issues for future work are as follows:(i)To develop strategies for obtaining more valuable results in related areas(ii)To apply these notions and results for studying fuzzy expert sets and applications in decision making problems.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.