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A soft set theoretic approach to an AG-groupoid via ideal theory with applications

Abstract

In this paper, we study the structural properties of a non-associative algebraic structure called an AG-groupoid by using soft set theory. We characterize a right regular class of an AG-groupoid in terms of soft intersection ideals and provide counter examples to discuss the converse part of various problems. We also characterize a weakly regular class of an AG***-groupoid by using generated ideals and soft intersection ideals. We investigate the relationship between SI-left-ideal, SI-right-ideal, SI-two-sided-ideal, and SI-interior-ideal of an AG-groupoid over a universe set by providing some practical examples.

Introduction

The concept of soft set theory was introduced by Molodtsov in [16]. This theory can be used as a generic mathematical tool for dealing with uncertainties. In soft set theory, the problem of setting the membership function does not arise, which makes the theory easily applied to many different fields [1, 2, 59]. At present, the research work on soft set theory in algebraic fields is progressing rapidly [19, 2123]. A soft set is a parameterized family of subsets of the universe set. In the real world, the parameters of this family arise from the view point of fuzzy set theory. Most of the researchers of algebraic structures have worked on the fuzzy aspect of soft sets. Soft set theory is applied in the field of optimization by Kovkov in [12]. Several similarity measures have been discussed in [15], decision-making problems have been studied in [21], and reduction of fuzzy soft sets and its applications in decision-making problems have been analyzed in [13]. The notions of soft numbers, soft derivatives, soft integrals, and many more have been formulated in [14]. This concept have been used for forecasting the export and import volumes in international trade [28]. A. Sezgin have introduced the concept of a soft sets in non-associative semigroups in [24] and studied soft intersection left (right, two-sided) ideals, (generalized) bi-ideals, interior ideals, and quasi-ideals in an AG-groupoid. A lot of work has been done on the applications of soft sets to a non-associative rings by T. Shah et al. in [25, 26]. They have characterized the non-associative rings through soft M-systems and different soft ideals to get generalized results.

This paper is the continuation of the work carried out by F. Yousafzai et al. in [29] in which they define the smallest one-sided ideals in an AG-groupoid and use them to characterize a strongly regular class of an AG-groupoid along with its semilattices and soft intersection left (right, two-sided) ideals, and bi-ideals. The main motivation behind this paper is to study some structural properties of a non-associative structure as it has not attracted much attention compared to associative structures. We investigate the notions of SI-left-ideal, SI -right-ideal, SI-two-sided-ideal, and SI -interior-ideal in an AG-groupoid. We provide examples/counter examples for these SI-ideals and study the relationship between them in detail. As an application of our results, we get characterizations of a right regular AG-groupoid and weakly regular AG***-groupoid in terms of SI-left-ideal, SI-right-ideal, SI-two-sided-ideal, and SI-interior-ideal.

AG-groupoids

An AG-groupoid is a non-associative and a non-commutative algebraic structure lying in a gray area between a groupoid and a commutative semigroup. Commutative law is given by abc=cba in ternary operations. By putting brackets on the left of this equation, i.e., (ab)c=(cb)a, in 1972, M. A. Kazim and M. Naseeruddin introduced a new algebraic structure called a left almost semigroup abbreviated as an LA-semigroup [10]. 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 [20].

This structure is closely related to a commutative semigroup because a commutative AG-groupoid is a semigroup [17]. It was proved in [10] that an AG-groupoid S is medial, that is, ab·cd=ac·bd holds for all a,b,c,dS. 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 [17]. In an AG-groupoid S with left identity, the paramedial law ab·cd=dc·ba holds for all a,b,c,dS. By using medial law with left identity, we get a·bc=b·ac for all a,b,cS. 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 [3, 4, 11, 17, 18, 20, 27] in this regard.

A nonempty subset A of an AG-groupoid S is called a left (right, interior) ideal of S if SAA (ASA,SA·SA). Equivalently, a nonempty subset A of an AG-groupoid S is called a left (right, interior) ideal of S if SAA (ASA,SA·SA). By two-sided ideal or simply ideal, we mean a nonempty subset of an AG-groupoid S which is both left and right ideal of S.

Soft sets

In [23], Sezgin and Atagun introduced some new operations on soft set theory and defined soft sets in the following way :

Let U be an initial universe set, E a set of parameters, P(U) the power set of U, and AE. Then, a soft set (briefly, a soft set) fA over U is a function defined by :

$$f_{A}:E\rightarrow P(U) \text{ such that } f_{A}(x)=\emptyset,\ \text{if}\ x\notin A. $$

Here, fA is called an approximate function. A soft set over U can be represented by the set of ordered pairs as follows:

$$f_{A}=\left \{ (x,\text{ }f_{A}(x)):\text{ }x\in E,\text{ }f_{A}(x)\in P(U)\right \}. $$

It is clear that a soft set is a parameterized family of subsets of U. The set of all soft sets is denoted by S(U).

Let fA,fBS(U). Then, fA is a soft subset of fB, denoted by \(f_{A}\overset {\sim }{\subseteq }f_{B}\) if fA(x)fB(x) for all xS. Two soft sets fA,fB are said to be equal soft sets if \(f_{A}\overset {\sim }{\subseteq }f_{B}\) and \(\overset { \sim }{f_{B}\subseteq f_{A}}\) and is denoted by \(f_{A}\overset {\sim }{=} f_{B} \). The union of fA and fB, denoted by \(f_{A}\overset {\sim }{ \cup }f_{B},\) is defined by \(f_{A}\overset {\sim }{\cup }f_{B}=f_{A\cup B},\) where fAB(x)=fA(x)fB(x),xE. In a similar way, we can define the intersection of fA and fB.

Let fA,fBS(U). Then, the soft product[23] of fA and fB, denoted by fAfB, is defined as follows :

$$(f_{A}\circ f_{B})(x)=\left \{ \begin{array}{c} \bigcup \limits_{x=yz}\{f_{A}(y)\cap g_{_{B}}(z)\} \text{ \ \ \ \ if } \exists \text{ }y,z\in S\text{ }\ni \text{ }x=yz \\ \emptyset \text{ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ otherwise } \end{array} \right.. $$

Let fA be a soft set of an AG-groupoid S over a universe U. Then, fA is called a soft intersection left ideal,right ideal,interior ideal (briefly,SI -left-ideal,SI-right-ideal, SI-interior-ideal) of S over U if it satisfies fA(xy)fA(y)(fA(xy)fA(x),fA(xy·z)fA(y)),x,yS. A soft set fA is called a soft intersection two-sided ideal (briefly, SI -two-sided-ideal) of S over U if fA is an SI -left-ideal and an SI-right-ideal of S over U.

Let A be a nonempty subset of S. We denote by XA the soft characteristic functionof A and define it as follows:

$$X_{A}=\left \{ \begin{array}{c} U\text{ \ \ \ if }x\in A \\ \emptyset \text{\ \ \ if }x\notin A\text{\ \ \ } \end{array} \right.. $$

Note that the soft characteristic mapping of the whole set S, denoted by XS, is called an identity soft mapping.

Basic results

Lemma 1

[29] For a nonempty subset A of an AG-groupoid S, the following conditions are equivalent :

(i)Ais a left ideal (right ideal,interior ideal)of S;

(ii)A soft set XAofSoverUis an SI-left-ideal (SI -right-ideal,SI-interior-ideal)ofSover U.

Lemma 2

[29] Let S be an AG-groupoid. For A,BS, the following assertions hold :

\((i) X_{A}\overset {\sim }{\cap }X_{B}=X_{A\cap B};\)

(ii)XAXB=XAB.

Remark 1

[29]The set (S (U),) forms an AG-groupoid and satisfies all the basic laws.

Remark 2

[29] If S is an AG-groupoid, then XSXS=XS.

Lemma 3

Let fA be anysoft set of a right regular AG-groupoid S with left identity over U. Then, fAisan SI-right-ideal (SI-left-ideal, SI-interior-ideal)ofSoverUif and only if fA=fAXS(fA=XSfA,fA=(XSfA)XS) and fAis soft semiprime.

Proof

It is simple. □

Lemma 4

For every SI-interior-ideal fA of a right regular AG-groupoid S with left identity over U,fA=XSfA=fXS.

Proof

Assume that fA is any SI-interior-ideal of S with left identity over U. Then, by using Remark 2 and Lemma 3, we have XSfA=(XSXS)fA=(fAXS)XS=(fAXS)(XSXS)=(XSXS)(XSfA)=((XSfA)XS)XS=fAXS and XSfA=(XSXS)fA=(fAXS)XS=(XSfA)XS=fA. □

Lemma 5

[29] Let fA be any soft set of an AG-groupoid S over U. Then, fAis anSI-right-ideal (SI-left-ideal)ofSoverUif and only if\(f_{A}\circ X_{S}\overset {\sim }{\subseteq }f_{A} (X_{S}\circ f_{A}\overset {\sim }{\subseteq }f_{A}).\)

Lemma 6

A right (left, two-sided) ideal R of an AG-groupoid S is semiprime if and only if XR is softsemiprime over U.

Proof

Let R be a right ideal of S. By Lemma 1, XR is an SI-right-ideal of S over U. If aS, then by given assumption (XR)(a)(XR)(a2). Now a2R, implies that aR. Thus every right ideal of S is semiprime. The converse is simple. Similarly every left or two-sided ideal of S is semiprime if and only if its identity soft mapping is soft semiprime over U. □

Corollary 1

If any SI-right-ideal (SI -left-ideal, SI-two-sided-ideal) of an AG-groupoid S is S-semiprime, then any right (left, two-sided) ideal of S is semiprime.

The converse of Lemma 6 is not true in general which can be followed from the following example.

Example 1

Let us consider an initial universe setUgiven by\(U= \mathbb {Z},\) and S={1,2,3,4,5}be a set of parameters with the following binary operation.

It is easy to check that (S,)is an AG-groupoid with left identity4.

Notice that the only left ideals ofSare {1,2,5}, {1,3,5}, {1,2,3,5}and {1,5}respectively which are semiprime. Clearly, the right and two-sided ideals ofSare {1,2,3,5}and {1,5}which are also semiprime. On the other hand,let A=Sand definea soft set fAofSoverUas follows:

\({f}_{A}{(x)=}\left \{ \begin {array}{c} \mathbb {Z} \ \text {if}\ x=1 \\ 4 \mathbb {Z} \ \text {if}\ x=2 \\ 4 \mathbb {Z} \ \text {if}\ x=3 \\ 8 \mathbb {Z} \ \text {if}\ x=4 \\ 2 \mathbb {Z} \ \text {if}\ x=5 \end {array} \right \} {.}\)

Then, fAis an SI-right-ideal (SI-left-ideal,SI-two-sided-ideal )ofSoverUbut fAis not soft semiprime. Indeed\(f_{A}(2) \varsupsetneq f_{A}(2^{2}).\)

Remark 3

If any SI-interior-ideal of an AG-groupoid S with left identity over U is an S-semiprime over U, then any interior ideal of S is semiprime. The converse inclusion is not true in general.

The following lemma will be used frequently in upcoming section without mention in the sequel.

Lemma 7

Let S be an AG-groupoid with left identity. Then, Sa and Sa2 are the left and interior ideals of S respectively.

Proof

It is simple. □

Right regular AG-groupoids

An element a of an AG-groupoid S is called a left (right) regular element of S if there exists some xS such that a=a2x (a=xa2) and S is called left (right) regular if every element of S is left (right) regular.

Remark 4

Let S be an AG-groupoid with left identity. Then, the concepts of left and right regularity coincide in S.

Indeed,for every aSthere exist some x,ySsuch that a=xa2=a2y. As a=xa2=ex·aa=aa·xe=a2y,and a=a2y=xa2also holds in a similar way.

Let us give an example of an AG-groupoid which will be used for the converse parts of various problems in this section.

Example 2

Let us consider an AG-groupoid S={1,2,3,4,5} with left identity 4 defined in the following multiplication table.

It is easy to check that S is non-commutative and non-associative.

An AG-groupoid S is called left (right) duo if every left (right) ideal of S is a two-sided ideal of S and is called duo if it is both left and right duo. Similarly an AG-groupoid S is called an SI-left (SI-right) duo if every SI-left-ideal (SI-right-ideal) of S over U is an SI-two-sided-ideal of S over U, and S is called an SI-duo if it is both an SI-left and an SI-right duo.

Lemma 8

If every SI-left-ideal of an AG-groupoid S with left identity over U is anSI-interior-ideal of S over U, then S is left duo.

Proof

Let I be any left ideal of S with left identity. Now by Lemma 1, the identity soft mapping XI is an SI-left-ideal of S over U. Thus, by hypothesis, XI is anSI-interior-ideal of S over U, and by using Lemma 1 again, I is an interior ideal of S. Thus IS=I·SS=S·IS=SS·IS=SI·SS=SI·SI. This shows that S is left duo. □

The converse part of Lemma 8 is not true in general. Let us consider an AG-groupoid S (from Example 2). It is easy to see that S is left duo because the only left ideal of S is {1,5} which is also a right ideal of S. Let A=S and define a soft set fA of S over U={p1,p2,p3,p4,p5,p6} as follows:

\(f_{A}(x)=\left \{ \begin {array}{c} U\ \text {if}\ x=1 \\ \{p_{1},p_{2},p_{3},p_{4}\}\ \text {if}x\ =2 \\ \{p_{2},p_{3},p_{4,},p_{5}\}\ \text {if}\ x=3 \\ \{p_{3},p_{4,},p_{5}\}\ \text {if}\ x=4 \\ \{p_{1},p_{2},p_{3},p_{4,}p_{5}\}\ \text {if}\ x=5 \end {array} \right \}.\)

Then, it is easy to see that fA is anSI -left-ideal of S over U but it is not anSI -interior-ideal of S over U because \(f_{A}(42\ast 4)\varsupsetneq f_{A}(2).\)

Corollary 2

Every interior ideal of an AG-groupoid S with left identity is a right ideal of S.

Theorem 1

Every SI-right-ideal of an AG-groupoid S with left identity is anSI-interior-ideal of S over U if and only if S is right duo.

Proof

It is simple. □

Theorem 2

Let S be a right regular AG-groupoid with left identity. Then, S is left duo if and only if every SI-left-ideal of S over U is anSI-interior-ideal of S over U.

Proof

Necessity. Let a right regular S with left identity be a left duo, and assume that fA is any SI-left-ideal of S over U. Let a,b,cS, then bxb2 for some xS. Since Sa is a left ideal of S, therefore by hypothesis, Sa is a two-sided ideal of S. Thus, ab·c=a(x·bbc=a(b·xbc=b(a·xbc=c(a·xbb. It follows that ab·cS(a·SSb(S·aS)b=(SS·aS)b=(Sa·SS)b(Sa·S)bSa·b. Thus, ab·c=ta·b for some tS, and therefore fA(ab·c)=fA(ta·b)fA(b), implies that fA is anSI-interior-ideal of S over U.

Sufficiency. It can be followed from Lemma 8. □

By left-right dual of above Theorem, we have the following Theorem :

Theorem 3

Let S be a right regular AG-groupoid with left identity. Then,S is right duo if and only if every SI-right-ideal of S over U is anSI-interior-ideal of S over U.

Lemma 9

A non-empty subset A of a right regular AG-groupoid S with left identity is a two-sided ideal of S if and only if it is an interior ideal of S.

Proof

It is simple. □

Lemma 10

Every left ideal of an AG-groupoid S with left identity is an interior ideal of S if S is an SI-left duo.

Proof

It can be followed from Lemmas 1 and 9. □

The converse of Lemma 10 is not true in general. The only left ideal of S (from Example 2) is {1,2} which is also an interior ideal of S. Let A={2,3,4,5} and define a soft set fA of S over \( U= \mathbb {Z} \) as follows:

\(f_{A}(x)=\left \{ \begin {array}{c} 4 \mathbb {Z} \ \text {if}\ x=2 \\ 8 \mathbb {Z} \ \text {if}\ x=3 \\ 16 \mathbb {Z} \ \text {if}\ x=4 \\ 2 \mathbb {Z} \ \text {if}\ x=5 \end {array} \right \} \).

Then, it is easy to see that fA is anSI -left-ideal of S over U but it is not anSI -right-ideal of S over U because \(f_{A}(2\ast 4)\varsupsetneq f_{A}(2).\)

It is easy to see that every SI-right-ideal of S with left identity over U is anSI-left-ideal of S over U.

Remark 5

Every SI-right-ideal of an AG-groupoid S with left identity is anSI-left-ideal of S over U, but the converse is not true in general.

Theorem 4

Every right ideal of an AG-groupoid S with left identity is an interior ideal of S if and only if S is an SI-right duo.

Proof

It is straightforward. □

Theorem 5

Let S be a right regular AG-groupoid with left identity. Then,S is an SI-left duo if and only if every left ideal of S is an interior ideal of S.

Proof

The direct part can be followed from Lemma 10. The converse is simple. □

By left-right dual of above Theorem, we have the following Theorem.

Theorem 6

Let S be a right regular AG-groupoid with left identity. Then,S is an SI-right duo if and only if every right ideal of S is an interior ideal of S.

Theorem 7

Let S be an AG-groupoid with left identity and E={xS:x=x2}S. Then the following assertions hold :

(i)Eforms a semilattice ;

(ii)Eis a singleton set,if a=ax·a,a,xS.

Proof

It is simple. □

Theorem 8

For an AG-groupoid S with left identity, the following conditions are equivalent :

(i)Sis right regular ;

(ii)For any interior idealIof S;

(a) II2,

(b)Iis semiprime.

(iii)For any SI-interior-ideal fAofSover U;

(a)\(f_{A}\overset {\sim }{\subseteq }f_{A}\circ f_{A},\)

(b) fAis soft semiprime over U.

(iv)Sis right regular and |E|=1, (a=ax·a, a,xE);

(v)Sis right regular and ESis semilattice.

Proof

(i)(v)(iv) can be followed from Theorem 7.

(iv)(iii):(a). Let fA be any SI -interior-ideal of a right regular S with left identity. Thus, for each aS, there exists some xS such that a=x·aa=a·xa=a·x(x·aa)=a·(ex)(a·xa)=a·(xa·a)(xe). Therefore,

$$\begin{array}{@{}rcl@{}} (f_{A}\circ f_{A})(a) &=&\bigcup \limits_{a=a\cdot (xa\cdot a)(xe)}\left \{ f_{A}(a)\cap f_{A}((xa\cdot a)(xe))\right \} \\ &\supseteq &f_{A}(a)\cap f_{A}((xa\cdot a)(xe))\supseteq f_{A}(a)\cap f_{A}(a)=f_{A}(a)\text{.} \end{array} $$

This shows that \(f_{A}\overset {\sim }{\subseteq }f_{A}\circ f_{A}.\)

(b). Also,

$$\begin{array}{@{}rcl@{}} a &=&x\cdot aa\leq ex\cdot aa=aa\cdot xe=(a\cdot xa^{2})(xe)=(x\cdot aa^{2})(xe)=x(ea\cdot aa)\cdot (xe) \\ &=&x(aa\cdot ae)\cdot (xe)=(aa)(x\cdot ae)\cdot (xe)=(ae\cdot x)(aa)\cdot (xe)=(ae\cdot x)a^{2}\cdot (xe). \end{array} $$

This implies that fA(a)=fA((ae·x)a2·(xe))fA(a2). Hence, fA is softsemiprime.

(iii)(ii):(a). Assume that I is any interior ideal of S, then by using Lemma 1, XI is an SI-interior-ideal of S over U. Let iI, then by using Lemma 2, we have XI(i)(XIXI)(i)=(XI)(i)=U. Hence, II2.

(b). Let i2I. Then, by given assumption, we have XI(i)XI(i2)=U. This implies that iI, and therefore I is semiprime.

(ii)(i): Let aS with left identity. Since Sa2 is an interior ideals of S, and clearly a2Sa2. Thus, by using given assumption, aSa2. Hence, S is right regular. □

Corollary 3

Every SI-interior-ideal of a right regular AG-groupoid S with left identity is softsemiprime over U.

Proof

Let fI be any SI-interior-ideal of a right regular S with left identity. Then, for each aS, there exists some xS such that fI(a)=fI(x·aa)=fI(a·xa)=fI(xa2·xa)fI(a2). □

Corollary 4

Let I be an interior ideal of an AG-groupoid S. Then, I is semiprime if and only if XI is softsemiprime over U.

Theorem 9

Let S be an AG-groupoid with left identity. Then, S is right regular if and only if every SI-interior-ideal fA of S over U is softidempotent and softsemiprime.

Proof

Necessity : Let fA be any SI-interior-ideal of a right regular S with left identity over U. Then, clearly \(f_{A}\circ f_{A} \overset {\sim }{\subseteq }f_{A}\). Now for each aS, there exists some xS such that a=x·aa=a·xa=ea·xa=ax·ae=(ae·x)a. Thus,

$$\begin{array}{@{}rcl@{}} (f_{A}\circ f_{A})(a) &=&\bigcup \limits_{a=(ae\cdot x)a}\left \{ f_{A}(ae\cdot x)\cap f_{A}(a)\right \} \supseteq f_{A}(ae\cdot x)\cap f_{A}(a) \\ &\supseteq &f_{A}(a)\cap f_{A}(a)=(f_{A}\cap f_{A})(a)=f_{A}(a)\text{.} \end{array} $$

This shows that fA is softidempotent over U. Again a=ex·aa=aa·xe=a2·xe. Therefore, fA(a)=fA(a2·xe)fA(a2). Hence, fA is softsemiprime over U.

Sufficiency : Since Sa2 is an interior ideal of S, therefore by Lemma 1, its soft characteristic function \(\phantom {\dot {i}\!}X_{Sa^{2}}\) is an SI-interior-ideal of S over U such that \(\phantom {\dot {i}\!}X_{Sa^{2}}\) is softidempotent over U. Since by given assumption, \(\phantom {\dot {i}\!}X_{Sa^{2}}\) is softsemiprime over U so by Corollary 4,Sa2 is semiprime. Since a2Sa2, therefore, aa2S. Thus, by using Lemma 2, we have \(\phantom {\dot {i}\!}X_{Sa^{2}}\circ X_{Sa^{2}}=X_{Sa^{2}},\) and \(\phantom {\dot {i}\!}X_{Sa^{2}}\circ X_{Sa^{2}}=X_{(Sa^{2}\cdot Sa^{2})}.\) Thus, we get \(\phantom {\dot {i}\!}X_{(Sa^{2}\cdot Sa^{2})}=X_{Sa^{2}}.\) This implies that \(\phantom {\dot {i}\!}X_{(Sa^{2}\cdot Sa^{2})}(a)=X_{Sa^{2}}(a)=U.\) Therefore, aSa2·Sa2=a2S·Sa2=(Sa2·S)a2Sa2. Hence,S is right regular. □

Lemma 11

Every SI-interior-ideal of a right regular AG-groupoid S with left identity over U is softidempotent.

Proof

Let fA be any SI-interior-ideal of a right regular S with left identity over U. Then, by using Lemma 4, \(f_{A}\circ f_{A}\overset {\sim }{\subseteq }f_{A}\). Since S right regular, therefore for every aS there exists some xS such that a=x·aa=a·xa=xa2·xa=ax·a2x=(a2x·x)a=(xx·aa)a=(aa·x2)a. Therefore,

$$\begin{array}{@{}rcl@{}} (f_{A}\circ f_{A})(a) &=&\bigcup \limits_{a=(aa\cdot x^{2})a}\left \{ f_{A}(aa\cdot x^{2})\cap f_{A}(a)\right \} \supseteq f_{A}(aa\cdot x^{2})\cap f_{A}(a) \\ &\supseteq &f_{A}(a)\cap f_{A}(a)=(f_{A}\cap f_{A})(a)\text{.} \end{array} $$

Thus, fAfA=fA. □

Theorem 10

Let S be an AG-groupoid with left identity and fA be any SI-interior-ideal of\(\ S\mathcal {\ }\)over U. Then,S is right regular if and only if fA=(XSfA)2 and fA is softsemiprime.

Proof

Necessity : Let fA be any SI-interior-ideal of a right regular S with left identity over U. Then, by using Lemmas 4 and 2, we have

$$(X_{S}\circ (X_{S}\circ f_{A}))\circ X_{S}=(X_{S}\circ f_{A})\circ X_{S}=(f_{A}\circ X_{S})\circ X_{S}=(X_{S}\circ X_{S})\circ f_{A}=X_{S}\circ f_{A}. $$

This shows that XSfA is anSI -interior-ideal of S over U. Now by using Lemmas 11 and 4, we have (XSfA)2=XSfA=fA. It is easy to see that fA is softsemiprime.

Sufficiency : Let fA=(XSfA)2 holds for any SI-interior-ideal fA of S over U. Then, by given assumption and Lemma 14, we get, \(f_{A}=(X_{S}\circ f_{A})^{2}=f_{A}^{2}.\) Thus, by using Theorem 9, S is right regular. □

Corollary 5

Let S be an AG-groupoid with left identity and fA be any SI-interior-ideal of\(\ S\mathcal {\ }\)over U. Then, S is right regular if and only if \(f_{A}=X_{S}\circ f_{A}^{2}\) and fA is soft semiprime.

Proof

From above theorem, \(f_{A}=(X_{S}\circ f_{A})^{2}=(X_{S}\circ f_{A})(X_{S}\circ f_{A})=(X_{S}\circ f_{A})\circ f_{A}=(f_{A}\circ f_{A})\circ X_{S}=(f_{A}\circ f_{A})\circ (X_{S}\circ X_{S})=(X_{S}\circ X_{S})\circ (f_{A}\circ f_{A})=X_{S}\circ f_{A}^{2}.\)

Lemma 12

Let S be an AG-groupoid with left identity and fA be any SI-left-ideal (SI -right-ideal,SI-two-sided-ideal)of\( \ S\mathcal {\ }\)over U. Then, S is right regular if and only if fA is softidempotent.

Proof

Necessity : Let fA be anSI-left-ideal of a right regular S with left identity over U. Then, it is easy to see that \( f_{A}\circ f_{A}\overset {\sim }{\subseteq }f_{A}.\) Let aS, then there exists xS such that a=aa·x=xa·a. Thus

$$(f_{A}\circ f_{A})(a)=\bigcup \limits_{a=xa\cdot a}\{f_{A}(xa)\cap f_{A}(a)\} \supseteq f_{A}(a)\cap f_{A}(a)=f_{A}(a), $$

which implies that fA is softidempotent.

Sufficiency : Assume that fAfA=fA holds for all SI-left-ideal of S with a left identity over U. Since Sa is a left ideal of S, therefore by Lemma 1, it follows that XSa is anSI-left-ideal of S over U. Since aSa, it follows that (XSa)(a)=U. By hypothesis and Lemma 2, we obtain (XSa)(XSa)=XSa and (XSa)(XSa)=XSa·Sa. Thus, we have (XSa·Sa)(a)=XSa(a)=U, which implies that aSa·Sa. Therefore, aSa·Sa=S2a2=Sa2. This shows that S is right regular. □

Theorem 11

Let S be an AG-groupoid with left identity and fA be any SI-left-ideal (SI -right-ideal,SI-two-sided-ideal) of\(\ S \mathcal {\ }\)over U. Then, S is right regular if and only if fA=(XSfA)(XSfA).

Proof

Necessity : Let S be a right regular S with left identity and let fA be any SI-left-ideal of S over U. It is easy to see that XSfA is also anSI -left-ideal of S over U. By Lemma 12, we obtain \((X_{S}\circ f_{A})\circ (X_{S}\circ f_{A})=(X_{S}\circ f_{A})\overset {\sim }{\subseteq } f_{A}.\) Let aS, then there exists xS such that a=aa·x=xa·a=(xa)(aa·x)=(xa)(xa·a). Therefore,

$$\begin{array}{@{}rcl@{}} \left((X_{S}\circ f_{A})\circ (X_{S}\circ f_{A})\right) (a) &\supseteq &(X_{S}\circ f_{A})(xa)\cap (X_{S}\circ f_{A})(xa\cdot a) \\ &\supseteq &X_{S}(x)\cap f_{A}(a)\cap X_{S}(xa)\cap f_{A}(a)=f_{A}(a), \end{array} $$

which is what we set out to prove.

Sufficiency : Suppose that fA=(XSfA)(XSfA) holds for all SI-left-ideal fA of S over U. Then \(f_{A}=(X_{S}\circ f_{A})\circ (X_{S}\circ f_{A})\overset {\sim }{ \subseteq }f_{A}\circ f_{A}\overset {\sim }{\subseteq }X_{S}\circ f_{A} \overset {\sim }{\subseteq }f_{A}.\) Thus, by Lemma 12, it follows that S is right regular. □

Lemma 13

Let fA be any SI-interior-ideal of a right regular AG-groupoid S with left identity\(\mathcal {\ }\)over U. Then, fA(a)=fA(a2), for all aS.

Proof

Let fA be any SI-interior-ideal of a right regular S with left identity over U. For aS, there exists some x in S such that a=ex·aa=aa·xe=(xe·a)a=(xe·a)(ex·aa)=(xe·a)(aa·xe)=aa·(xe·a)(xe)=ea2·(xe·a)(xe). Therefore fA(a)=fA(ea2·(xe·a)(xe))fA(a2)=fA(aa)=fA(a(ex·aa))=fA(a(aa·xe))=fA((aa)(a·xe))=fA((xe·a)(aa))fA(a). That is, fA(a)=fA(a2),aS

The converse of Lemma 13 is not true in general. Let us consider an AG-groupoid S (from Example 2). Let A={1,2,4,5} and define a soft set fA of S over \(U=\left \{ \left [ \begin {array}{cc} 0 & 0 \\ x & x \end {array} \right ] /x\in \mathbb {Z} _{3}\right \} (\)the set of all 2×2 matrices with entries from \( \mathbb {Z} _{3})\) as follows:

\(f_{A}(x)=\left \{ \begin {array}{c} \left \{ \left [ \begin {array}{cc} 0 & 0 \\ 0 & 0 \end {array} \right ],\left [ \begin {array}{cc} 0 & 0 \\ 1 & 1 \end {array} \right ],\left [ \begin {array}{cc} 0 & 0 \\ 2 & 2 \end {array} \right ] \right \}\ \text {if}\ x=1 \\ \left \{ \left [ \begin {array}{cc} 0 & 0 \\ 1 & 1 \end {array} \right ],\left [ \begin {array}{cc} 0 & 0 \\ 2 & 2 \end {array} \right ] \right \}\ \text {if}\ x=2 \\ \left \{ \left [ \begin {array}{cc} 0 & 0 \\ 2 & 2 \end {array} \right ] \right \}\ \text {if}\ x=4 \\ \left \{ \left [ \begin {array}{cc} 0 & 0 \\ 1 & 1 \end {array} \right ],\left [ \begin {array}{cc} 0 & 0 \\ 2 & 2 \end {array} \right ] \right \}\ \text {if}\ x=\mathit {5} \end {array} \right \}.\)

It is easy to see that fA is anSI -interior-ideal of S such that fA(x)fA(x2),xS but S is not right regular.

On the other hand, it is easy to see that every SI -two-sided-ideal of S over U is anSI -interior-ideal of S over U.

Remark 6

Every SI-two-sided-ideal of a right regular AG-groupoid S with left identity\(\mathcal {\ }\)over U is an SI-interior-ideal of S over U but the converse is not true in general.

Theorem 12

For an AG-groupoid S with left identity, the following conditions are equivalent :

(i)Sis right regular ;

(ii)Every interior ideal ofSis semiprime ;

(iii)Every SI-interior-ideal ofSoverUis soft semiprime ;

(iv)For every SI-interior-ideal fAofSover U, fA(a)=fA(a2), aS.

Proof

(i)(iv) can be followed from Lemma 13.

(iv)(iii) and (iii)(ii) are obvious.

(ii)(i): Since Sa2 is an interior ideal of S with left identity such that a2Sa2, therefore by given assumption, we have aSa2. Thus, S is right regular. □

Weakly regular AG ***-groupoids

An AG-groupoid S is called an AG ***-groupoid [29] if the following conditions are satisfied :

(i) For all a,b,cS,a·bc=b·ac;

(ii) For all aS, there exist some b,cS such that a=bc.

An AG-groupoid satisfying (i) is called an AG **-groupoid. The condition (ii) for an AG **-groupoid to become an AG ***-groupoid is equivalent to S=S2.

Let S={1,2,3,4} be an AG-groupoid define in the following multiplication table.

It is easy to verify that (S,·) is an AG *** -groupoid.

Note that every AG-groupoid with left identity is an AG *** -groupoid but the converse is not true in general. An AG-groupoid in the above example is an AG ***-groupoid, but it does not contains a left identity. Hence, we can say that an AG ***-groupoid is the generalization of an AG-groupoid with left identity.

An element a of an AG-groupoid S is called a weakly regular element of S if there exist some x,yS such that a=ax·ay and S is called weakly regular if every element of S is weakly regular.

Remark 7

Let S be anAG ***-groupoid. Then, the concepts of weak and right regularity coincide in S.

Let S be an AG ***-groupoid. From now onward,R (resp.L) will denote any right (resp. left) ideal of \(S; \left \langle R\right \rangle _{a^{2}}\) will denote a right ideal Sa2a2 of S containing a2 and 〈La will denote a left ideal Saa of S containing a;fA(resp.gB) will denote any SI-right-ideal of S (resp. SI-left-ideal of S) over U unless otherwise specified.

Theorem 13

LetSbe anAG ***-groupoid. Then,Sis weakly regular if and only if \(\left \langle R\right \rangle _{a^{2}}\cap \left \langle L\right \rangle _{a}=\left \langle R\right \rangle _{a^{2}}^{2}\left \langle L\right \rangle _{a}^{2}\) and \(\left \langle R\right \rangle _{a^{2}}\) is semiprime.

Proof

Necessity : Let S be weakly regular. It is easy to see that \( \left \langle R\right \rangle _{a^{2}}^{2}\left \langle L\right \rangle _{a}^{2}\subseteq \left \langle R\right \rangle _{a^{2}}\cap \left \langle L\right \rangle _{a}.\) Let \(a\in \left \langle R\right \rangle _{a^{2}}\cap \left \langle L\right \rangle _{a}.\) Then, there exist some x,yS such that

$$\begin{array}{@{}rcl@{}} a &=&ax\cdot ay=(ax\cdot ay)x\cdot (ax\cdot ay)y=(x\cdot ay)(ax)\cdot (y\cdot ay)(ax) \\ &=&(a\cdot xy)(ax)\cdot (ay^{2})(ax)=(a\cdot xy)(ax)\cdot (xa)(y^{2}a) \\ &\in &(\left \langle R\right \rangle_{a^{2}}S\cdot \left \langle R\right \rangle_{a^{2}}S)(S\left \langle L\right \rangle_{a}\cdot S\left \langle L\right \rangle_{a})\subseteq \left \langle R\right \rangle_{a^{2}}^{2}\left \langle L\right \rangle_{a}^{2}, \end{array} $$

which shows that \(\left \langle R\right \rangle _{a^{2}}\cap \left \langle L\right \rangle _{a}=\left \langle R\right \rangle _{a^{2}}^{2}\left \langle L\right \rangle _{a}^{2}.\) It is easy to see that \(\left \langle R\right \rangle _{a^{2}}\) is semiprime.

Sufficiency : Since Sa2a2 and Saa are the right and left ideals of S containing a2 and a respectively. Thus, by using given assumption, we get

$$\begin{array}{@{}rcl@{}} a &\in &\left(Sa^{2}\cup a^{2}\right)\cap (Sa\cup a)=\left(Sa^{2}\cup a^{2}\right)^{2}(Sa\cup a)^{2} \\ &=&\left(Sa^{2}\cup a^{2}\right)\left(Sa^{2}\cup a\right)\cdot (Sa\cup a)(Sa\cup a) \\ &\subseteq &S\left(Sa^{2}\cup a\right)\cdot S(Sa\cup a)=\left(S\cdot Sa^{2}\cup Sa\right)(S\cdot Sa\cup Sa) \\ &=&\left(a^{2}S\cdot S\cup Sa\right)(aS\cdot S\cup Sa)=\left(Sa^{2}\cup Sa\right)(Sa\cup Sa) \\ &=&\left(a^{2}S\cup Sa\right)(Sa\cup Sa)=(Sa\cdot a\cup Sa)(Sa\cup Sa) \\ &\subseteq &(Sa\cup Sa)(Sa\cup Sa)=Sa\cdot Sa=aS\cdot aS. \end{array} $$

This implies that S is weakly regular. □

Corollary 6

LetSbe anAG ***-groupoid. Then,Sis weakly regular if and only if \( \left \langle R\right \rangle _{a^{2}}\cap \left \langle L\right \rangle _{a}=\left \langle L\right \rangle _{a}^{2}\left \langle R\right \rangle _{a^{2}}^{2}\) and \(\left \langle R\right \rangle _{a^{2}}\) is semiprime.

Theorem 14

LetSbe anAG ***-groupoid. Then, the following conditions are equivalent :

(i)Sis weakly regular ;

\((ii) \left \langle R\right \rangle _{a^{2}}\cap \left \langle L\right \rangle _{a}=\left \langle L\right \rangle _{a}^{2}\left \langle R\right \rangle _{a^{2}}^{2}\)and\(\left \langle R\right \rangle _{a^{2}}\)is semiprime ;

(iii) RL=L2R2 andRsemiprime ;

\(\left (iv\right) \ f_{A}\overset {\sim }{\cap }g_{B}=(f_{A}\circ g_{B})\circ (f_{A}\circ g_{B})\ \)and fAis soft semiprime ;

(v)Sisweakly regular and |E|=1,(a=ax·a,a,xE);

(vi)Sisweakly regular and ESis semilattice.

Proof

(i)(vi)(v): It can be followed from Theorem 7.

(v)(iv): Let fA and gB be any SI -right-ideal and SI-left-ideal of a weakly regular S over U respectively. From Lemma 5, it is easy to show that \((f_{A}\circ g_{B})\circ (f_{A}\circ g_{B})\overset {\sim }{\subseteq }f_{A}\overset {\sim } {\cap }g_{B}.\) Now for aS, there exist some x,yS such that

$$\begin{array}{@{}rcl@{}} a &=&ax\cdot ay=(ax\cdot ay)x\cdot (ax\cdot ay)y=(ax\cdot ay)\cdot ((ax\cdot ay)x)y \\ &=&(ax\cdot ay)\cdot (yx)(ax\cdot ay)=(ax\cdot ay)\cdot (ax)(yx\cdot ay) \\ &=&(ax\cdot ay)\cdot (ay\cdot yx)(xa)=(ax\cdot ay)\cdot ((yx\cdot y)a)(xa) \\ &=&(ax)((yx\cdot y)a)\cdot (ay)(xa)=(ax)(ba)\cdot (ay)(xa),\ \text{where}\ yx\cdot y=b. \end{array} $$

Therefore,

$$\begin{array}{@{}rcl@{}} ((f_{A}\circ g_{B})\circ (f_{A}\circ g_{B}))(a) &=&\bigcup \limits_{a=(ax)(ba)\cdot (ay)(xa)}\{(f_{A}\circ g_{B})(ax\cdot ba) \\ &&\cap (f_{A}\circ g_{B})(ay\cdot xa)\} \\ &\supseteq &\bigcup \limits_{ax\cdot ba=ax\cdot ba}\{f_{A}(ax)\cap g_{B}(ba)\} \\ &&\cap \bigcup \limits_{ay\cdot xa=ay\cdot xa}\{f_{A}(ay)\cap g_{B}(xa)\} \\ &\supseteq &f_{A}(ax)\cap g_{B}(ba)\cap f_{A}(ay)\cap g_{B}(xa) \\ &\supseteq &f_{A}(a)\cap g_{B}(a), \end{array} $$

which shows that \((f_{A}\circ g_{B})\circ (f_{A}\circ g_{B})\overset {\sim }{ \supseteq }f_{A}\overset {\sim }{\cap }g_{B}.\) Hence, \(f_{A}\overset {\sim }{ \cap }g_{B}=(f_{A}\circ g_{B})\circ (f_{A}\circ g_{B})\). Also by using Lemma 3, fA is softsemiprime.

(iv)(iii): Let R and L be any left and right ideals of S. Then, by using Lemma 1, XR and XL are the SI-right-ideal and SI-left-ideal of S over U respectively. Now by using Lemma 2, we get \(X_{R\cap L}=X_{R}\overset {\sim }{\cap }X_{L}=(X_{R}\circ X_{L})\circ (X_{R}\circ X_{L})=(X_{R}\circ X_{R})\circ (X_{L}\circ X_{L})=X_{R^{2}}\circ X_{L^{2}}=X_{R^{2}L^{2}}=X_{L^{2}R^{2}},\)which implies that RL=L2R2.

(iii)(ii): It is simple.

(ii)(i): It can be followed from Corollary 6. □

Lemma 14

Let R be a right ideal and L be a left ideal of a unitary AG-groupoid S with left identity respectively. Then,RL is a left ideal of S.

Proof

It is simple. □

Theorem 15

LetSbe anAG ***-groupoid. Then, the following conditions are equivalent :

(i)Sis weakly regular ;

(ii)\(\left \langle R\right \rangle _{a^{2}}\cap \left \langle L\right \rangle _{a}=\left \langle R\right \rangle _{a^{2}}\left \langle L\right \rangle _{a}\cdot \left \langle R\right \rangle _{a^{2}}\)and\(\left \langle R\right \rangle _{a^{2}}\)is semiprime ;

(iii) RL=RL·RandRis semiprime ;

(iv)\(f_{A}\overset {\sim }{\cap }g_{B}=(f_{A}\circ g_{B})\circ f_{A}\mathit {\ }\)and fAis soft semiprime ;

(v)Sisweakly regular and |E|=1,(a=ax·a,a,xE);

(vi)Sisweakly regular and ESis semilattice.

Proof

(i)(vi)(v): It can be followed from Theorem 7.

(v)(iv): Let fA and gB be any SI -left-ideals of a weakly regular S over U. Now, for aS, there exist some x,yS such that a=ax·ay=ax·(ax·ay)y=((ax·ay)y·x)a=(xy·(ax·ay))a=(ax·(xy·ay))a=(ax·(a·(xy)y))a.

Therefore,

$$\begin{array}{@{}rcl@{}} ((f_{A}\circ g_{B})\circ f_{A})(a) &=&\bigcup \limits_{a=(ax\cdot (a\cdot (xy)y))a}\{(f_{A}\circ g_{B})(ax\cdot (a\cdot (xy)y))\cap g_{B}(a)\} \\ &\supseteq &\bigcup \limits_{ax\cdot (a\cdot (xy)y=ax\cdot (a\cdot (xy)y}\{f_{A}(ax)\cap g_{B}(a\cdot (xy)y)\} \cap g_{B}(a) \\ &\supseteq &f_{A}(ax)\cap g_{B}(a\cdot (xy)y)\cap g_{B}(a)\supseteq f_{A}(a)\cap g_{B}(a)\text{,} \end{array} $$

which shows that \((f_{A}\circ g_{B})\circ f_{A}\overset {\sim }{\supseteq } f_{A}\overset {\sim }{\cap }g_{B}\). By using Lemmas 5 and 3, it is easy to show that \((f_{A}\circ g_{B})\circ f_{A}\overset {\sim }{ \subseteq }f_{A}\overset {\sim }{\cap }g_{B}.\) Thus, \(f_{A}\overset {\sim }{ \cap }g_{B}=(f_{A}\circ g_{B})\circ f_{A}\). Also, by using Lemma 3, fA is softsemiprime.

(iv)(iii): Let R and L be any left and right ideals of S respectively. Then, by Lemma 1, XR and XL are the SI-right-ideal and SI -left-ideal of S over U respectively. Now, by using Lemmas 2, 14, we get \(X_{R\cap L}=X_{R}\overset {\sim }{\cap } X_{L}=(X_{R}\circ X_{L})\circ X_{L}=X_{RL\cdot R},\) which shows that RL=RL·R. Also, by using Lemma 6, R is semiprime.

(iii)(ii): It is obvious.

(ii)(i): Since Sa2a2 and Saa are the right and left ideals of S containing a2 and a respectively. Thus, by using given assumption and Lemma, we get

$$\begin{array}{@{}rcl@{}} a &\in &(Sa^{2}\cup a^{2})\cap (Sa\cup a)=(Sa^{2}\cup a^{2})(Sa\cup a)\cdot (Sa^{2}\cup a^{2}) \\ &\subseteq &S(Sa\cup a)\cdot (Sa^{2}\cup a^{2})=(S^{2}a\cup Sa)(Sa^{2}\cup a^{2}) \\ &=&(S^{2}a\cdot Sa^{2})\cup (S^{2}a\cdot a^{2})\cup (Sa\cdot Sa^{2})\cup (S^{2}a\cdot a^{2}) \\ &\subseteq &(Sa\cdot a^{2}S)\cup (Sa\cdot Sa)\cup (Sa\cdot a^{2}S)\cup (Sa\cdot Sa) \\ &\subseteq &(Sa\cdot Sa)\cup (Sa\cdot Sa)\cup (Sa\cdot Sa)\cup (Sa\cdot Sa) \\ &=&Sa\cdot Sa=aS\cdot aS. \end{array} $$

Hence, S is weakly regular. □

Comparison of SI-left (right, two-sided, interior) ideals

A very major and an abstract conclusion from this section is that SI-left-ideal, SI-right-ideal and SI-interior-ideal need not to be coincide in an AG-groupoid S even if S has a left identity, but they will coincide in a right regular class of an AG-groupoid S with left identity.

E-1. Take a collection of 8 chemicals as an initial universe set U given by U={s1,s2,s3,s4,s5,s6,s7,s8}.

Let a set of parameters S={1,2,3,4,5} be a set of particular properties of each chemical in U with the following type of natures :

1 stands for the parameter "density",

2 stands for the parameter "melting point",

3 stands for the parameter "combustion",

4 stands for the parameter "enthalpy",

5 stands for the parameter "toxicity".

Let us define the following binary operation on a set of parameters S as follows.

It is easy to check that (S,) is non-commutative and non-associative. Also, by routine calculation, one can easily verify that (S,) forms an AG-groupoid with left identity 4. Note that S is left (right) regular. Indeed, for aS there does exists some xS such that a=xa2(a=a2x).

Let A=S and define a soft set fA of S over U as follows :

\(f_{A}(x)=\left \{ \begin {array}{c} \{s_{1},s_{2},s_{3},s_{4,}s_{5},s_{6}\}\ \text {if}\ x=1 \\ \{s_{2},s_{3},s_{4,}\}\ \text {if}\ x=2 \\ \{s_{2},s_{3}\ \text {if}\ x=3=4=5 \end {array} \right \}.\)

Then, it is easy to verify that fA is an SI -interior-ideal of S over U.

E-2. There are seven civil engineers in an initial universe set U given by U={s1,s2,s3,s4,s5,s6,s7}.

Let a set of parameters S={1,2,3} be a set of status of each civil engineer in U with the following type of attributes:

1 stands for the parameter “critical thinking”,

2 stands for the parameter “decision making”,

3 stands for the parameter “project management”.

Let us define the following binary operation on a set of parameters S as follows.

It is easy to check that (S,) is non-commutative and non-associative. One can easily verify that (S,) forms an AG-groupoid. Note that S is not left (right) regular. Indeed for 3S there does not exists some xS such that 3=x32(3=32x). Let A=S and define a soft set fA of S over U as follows :

\(f_{A}(x)=\left \{ \begin {array}{c} \{s_{1},s_{2},s_{3},s_{4}\}\ \text {if}\ x=1 \\ \{s_{1},s_{2},s_{3}\}\ \text {if}\ x=2 \\ \{s_{2},s_{3}\}\ \text {if}\ x=3 \end {array} \right \}.\)

Then, it is easy to verify that fA is an SI -interior-ideal of S over U but it is not an SI -left-ideal, SI-right-ideal, and SI -interior-ideal of S which can be seen from the following :

$$f_{A}(2\ast 2)\varsupsetneq f_{A}(2)\text{ and\textit{\ }}f_{A}(3\ast 2)\varsupsetneq f(2). $$

Lemma 15

Every SI-right-ideal of an AG-groupoid S with left identity over U is an SI-left-ideal of S over U.

Proof

It is simple. □

The converse of above Lemma is not true in general which can be seen from the following example.

E-3. Let us consider an AG-groupoid S with left identity 4 given in an Example 1 with an initial universe set U={s1,s2,...,s12}. Let a set of parameters S={1,2,3,4,5} be a set of status of houses in which,

1 stands for the parameter “beautiful”,

2 stands for the parameter “cheap”,

3 stands for the parameter “in good location”,

4 stands for the parameter “in green surroundings”,

5 stands for the parameter “secure”.

It is important to note that S is not right regular because for 3S there does not exists some xS such that 3=x32.

Let A=S and define a soft set fA of S over U as follows :

\(f_{A}(x)=\left \{ \begin {array}{c} U\ \text {if}\ x=1 \\ \{s_{2},s_{3},s_{4,}s_{5,}s_{6,}s_{7,}s_{8}\}\ \text {if}\ x=2 \\ \{s_{2},s_{3},s_{4,}s_{5,}s_{6}\}\ \text {if}\ x=3 \\ \{s_{2},s_{3},s_{4,}s_{5}\}\ \text {if}\ x=4 \\ \{s_{1},s_{2},s_{3},s_{4,}s_{5,}s_{6,}s_{7,}s_{8},s_{9},s_{10}\}\ \text {if}\ x=5 \end {array} \right \} \).

It is easy to verify that fA is an SI-left-ideal of S over U, but it is not an SI-right-ideal of S over U, because \(f_{A}(2\ast 4)\varsupsetneq f_{A}(2).\) Also, one can easily see that fA is an SI-interior-ideal of S over U but it is not an SI-two-sided-ideal of S over U.

Note that every SI-two-sided-ideal of an AG-groupoid S with left identity over U is an SI-interior-ideal of S over U.

Theorem 16

Let fA be any soft set of a right regular AG-groupoid S with left identity over U. Then, fAisanSI -left-idealofSoverUif and only if fAisanSI-right-idealofSoverUif and only if fAisanSI-two-sided-idealofSoverUif and only if fAisanSI -interior-idealofSover U.

Proof

Assume that fA is any SI-left-ideal of a right regular S with left identity over U. Let a,bS. For aS, there exists some xS such that a=xa2. Thus, ab=xa2·b=(a·xa)b=(b·xa)a. Therefore, fA((b·xa)a)fA(a). Now, by using Lemma 15, fAis an SI-left-ideal of S over U if and only if fA is an SI-right-ideal of S over U. Let fA is any SI-right-ideal of a right regular with left identity over U. Let a,b,cS, then fA(ab·c)=fA((xa2·b)c)=fA(cb·xa2)=fA(a2x·bc)=fA(b(a2x·c))fA(b). Again assume that fA is any SI -interior-ideal of a right regular S with left identity over U. Thus, fA(ab)fA(xa2·b)fA(a2)=fA(xa2·xa2)=fA(a2x·a2x)=fA((aa)(a2x·x))fA(a), which is what we set out to prove. □

Conclusions

Every AG-groupoid with left identity can be considered as an AG***-groupoid, but the converse is not true in general. This leads us to the fact that an AG***-groupoid can be seen as the generalization of an AG-groupoid with left identity. Thus, the results of “Right regular AG-groupoids” section can be trivially followed for an AG***-groupoid.

The idea of soft sets in an AG-groupoid will help us in verifying the existing characterizations and to achieving new and generalized results in future works. Some of them are as under:

1. To generalize the results of a semigroups using soft sets.

2. To characterize a newly developed substructure called an AG***-groupoid through soft sets.

3. To study the structural properties of an AG-hypergroupoid by using soft sets.

4. To introduce and examine the concept of a Γ-AG-groupoid in terms of soft sets.

Availability of data and materials

No data were used to support this study.

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Acknowledgements

The authors are grateful to the reviewers’ valuable comments that improved the manuscript.

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Yousafzai, F., Khalaf, M.M. A soft set theoretic approach to an AG-groupoid via ideal theory with applications. J Egypt Math Soc 27, 58 (2019). https://doi.org/10.1186/s42787-019-0060-7

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Keywords

  • Left invertive law
  • Soft-sets
  • AG-groupoid
  • Right regularity
  • Weak regularity and SI-ideals