Fuzzy ∗–ideals and their applications in characterizing abundance and regularity of a semigroup

1 Introduction

It is well known that a fuzzy subset f of a non-empty S is an arbitrary mapping of S into the closed interval [0, 1]. In group theory, Rosenfeld (1971) [17] explained the definition of a fuzzy subgroup of a group by exploiting the fuzzy subset. In contrast to the fuzzy group, many authors (see [1, 18]) characterized fuzzy subsemigroups, fuzzy ideals and fuzzy relations of a semigroup, respectively. For example, Kehayopulu and Tsingelis [8] defined the fuzzy Green’s relations in ordered groupoids, and obtained the decompositions of ordered semigroups; Bashir, Abbas and Mazhar [1] studied fuzzy ideals with three-dimensional congruence relation on semigroups; Cristea, Mahboob and Khan [2] considered a new type of fuzzy quasi-ideals of ordered semigroups; Li, Xu and Huang [10] characterized bipolar fuzzy abundant semigroups. Recently, Pal [15] and Rao [16] studied completely prime fuzzy ideals and fuzzy quasi-interior ideals of semigroups, respectively. In this paper, we shall give another characterization of fuzzy ideals of an arbitrary semigroup.

For any two elements a, b of a semigroup S, we say that a, b are L - related in S if there exist x, y ∈ S¹ satisfying a = xb and b = ya. The relation R on an arbitrary semigroup S can be defined dually. A semigroup S is regular if for all a ∈ S there is x ∈ S with a = axa . Accordingly, two elements a, b of a semigroup S are L * - related [resp., R * - related] if and only if there exist x, y ∈ S¹ such that ax = ay ⇔ bx = by [resp., xa = ya ⇔ xb = yb]; It is clear that the relations L * and R * are generalized Green’s L and R , respectively. For the convenience of our statement, the L * - class [resp., R * - class] containing the element a of S will be written by L * a [resp., R * a ]; The idempotent in R * a [resp., L * a ] is denoted by a⁺ [resp., a^*]; E (S) denotes the set of idempotents of S; A semigroup S is abundant if E ( S ) ⋂ L * a ≠ ∅ and E ( S ) ⋂ R * a ≠ ∅ for all a ∈ S .

According to Fountain [4], a left [resp., right] ideal I of an arbitrary semigroup S is a left [resp., right] *–ideal of S if for all a ∈ I, L * a ⊆ I [resp., R * a ⊆ I ]. In particular, a subset I of S is said to be a *–ideal of S if it is both a left *–ideal and a right *–ideal of S . Usually, the smallest left *–ideal [resp., right *–ideal, *–ideal] containing the element a of a semigroup S will be denoted by L * ( a ) [resp., R * ( a ) , J * ( a ) ]. If e ∈ E (S) , then Se and eS are the smallest left and right *–ideal containing e, respectively. It can be easily seen that a semigroup S is abundant if and only if for each a ∈ S, there exist e, f ∈ E (S) satisfying L * ( a ) = Se and R * ( a ) = fS .

Inspired by the research of fuzzy ideals in ordered semigroups and rings, and motivated by Mordeson, Malik and Kuroki’s work [13] in (regular) semigroups in the light of fuzzy subsets, we [10] gave some new fuzzy relations (i.e., bipolar fuzzy relations) on an arbitrary semigroup to describe abundance of a semigroup. It is an interesting thing to elaborate the abundance of a semigroup. Fountain and others (see, [3, 12])) introduced numerous classes of semigroups by imposing conditions on abundance of a semigroup, and studied many interesting properties of such semigroups. As a continuousness of these works, this paper shall give the notion of fuzzy *–ideals of a semigroup to characterize abundance of an arbitrary semigroup.

The contents are arranged as follows. Section 2 recalls some basic concepts and some known results. Section 3 proposes notions of fuzzy left (right)*–ideals and fuzzy *–ideals of a semigroup and give some constructions. The main result of this paper is presented in section 4, that is, sufficient and necessary conditions for an arbitrary semigroup to be abundant are given by using fuzzy *–ideals of a semigroup. In section 5, some applications are obtained. In the last section, we give some remarks.

2 Preliminaries

First, we state some known results and notations (see, [4, 13]).

Lemma 2.1. [4] Let S be a semigroup and a, b ∈ S, e ∈ E (S). Then a L * e [ resp . , a R * e ] if and only if ae = a [resp . , ea = a] and for all x, y ∈ S¹, ax = ay [resp . , xa = ya] implies ex = ey [resp . , xe = ye]. In particular, L * and R * are a right congruence and a left congruence on S, respectively.

We observe here that L ⊆ L * and R ⊆ R * from Lemma 2.1. However, for regular elements a, b, we also note that a L * b [resp., a R * b ] if and only if a L b [resp., a R b ]. Next, we recall some basic concepts in the fuzzy set theory.

Suppose that T be a subset of a non-empty set S . The characteristic function of T is written by C _T and defined by C _T  (x) =1 if x ∈ T and C _T  (x) =0 if x ∈ S \ T . Let f and g be two fuzzy subsets of a semigroup S . Then f ⊆ g [resp., f ∩ g] is defined by f (x) ≤ g (x) [resp., (f ∩ g) (x) = f (x) ∧ g (x)] for all x ∈ S, and the product f ∘ g is given by

f ∘ g ( x ) = { ∨ x = yz { f ( y ) ∧ g ( z ) } ( ∃ y , z ∈ S ) x = yz 0 otherwise(1)

As is well known, the operation “ ∘ " is associative. A fuzzy subset f of a semigroup S is said to be a fuzzy left [resp., right] ideal of S if f (ab) ≥ f (b) [resp., f (ab) ≥ f (a)] for all a, b ∈ S . In particular, f is called a fuzzy ideal of S if it is both a fuzzy left ideal and a fuzzy right ideal of S .

Lemma 2.2. [13] Let f, g be a fuzzy left ideal and a fuzzy right ideal of a semigroup S, respectively. Then, for all a, b ∈ S, a L b [resp., a R b ] if and only if f (a) = f (b) [resp., g (a) = g (b)].

In this section, we give definitions of a fuzzy left (right)*–ideal, and fuzzy *–ideal of a semigroup. Especially, some constructions of such ideals are obtained.

Definition 3.1. Let f be a fuzzy subset of a semigroup S . Then f is called a fuzzy left [resp., right] *–ideal of S if it satisfies:

(i) for all a, b ∈ S, f (ab) ≥ f (b) [resp., f (ab) ≥ f (a)];

(ii) for any a ∈ S, L * a ∩ E ( S ) ≠ ∅ [resp., R * a ∩ E ( S ) ≠ ∅ ] implies that f (a^*) ≥ f (a) [resp., f (a⁺) ≥ f (a)], where a^* [resp., a⁺] is any element of L * a ∩ E ( S ) [resp., R * a ∩ E ( S ) ].

Especially, we say that f is a fuzzy *–ideal of S if it is both a fuzzy left *–ideal and a fuzzy right *–ideal of S .

From Definition 3.1, it is easy to check that, in an arbitrary regular semigroup S, a fuzzy left (right) ideal of S is a fuzzy left (right)*–ideal of S. Now, we give two non-regular semigroups satisfying the conditions of Definition 3.1.

It is easy to see that S is a non-regular abundant semigroup with E (S) = { 0,  e,  h } . In fact, it is routine to check that the L * classes of S are {0} , {e} , {h, a} , and that the R * classes of S are {0} , {h} , {e, a} . Moreover, S is non-regular since a ≠ axa for all x ∈ S . Consider two fuzzy subsets of S as follows: f : S ⟶ [ 0 , 1 ] , f ( 0 ) = 1 , f ( e ) = 1 / 2 , f ( h ) = f ( a ) = 1 / 3 ; g : S ⟶ [ 0 , 1 ] , g ( 0 ) = 1 , g ( h ) = 1 / 3 , g ( e ) = g ( a ) = 1 / 2 . It is routine to check that f [resp., g] is a fuzzy left [resp., right] *–ideal of S .

Example 3.2. Let ℕ be the set of all non-negative integers and S = { 1 2 n | n ∈ ℕ } . Then, it is easy to check that S is a non-regular semigroup with respect to a general multiplication, and that for all a ∈ S, a^* = a⁺ = 1 . Consider a fuzzy subset g of S as follows: g : S ⟶ [ 0 , 1 ] , g ( a ) = 1 - a * . It is easy to check that g is a fuzzy *–ideal of S .

Remark 3.1. A fuzzy left (right)*–ideal of a semigroup S is a fuzzy left (right) ideal of S, and the converse is not true. In fact, in Example 3.2, consider a fuzzy subset f of S as follows: f : S ⟶ [ 0 , 1 ] , f ( a ) = 1 - a Obviously, f is a fuzzy ideal of S. However, f is not a fuzzy *–ideal of S since f (a) >0 = f (a^*) for all a ∈ S \ {1} .

Now, we give some constructions of fuzzy left (right) *–ideals of an arbitrary semigroup.

Theorem 3.2. Let T be a left [resp., right] *–ideal of an arbitrary semigroup S. Then C _T is a fuzzy left [resp., right] *–ideal of S .

Proof. Let T be a left *–ideal of S . Then it is easy to see that C _T is a fuzzy left ideal of S . Suppose that a ∈ S and L * a ∩ E ( S ) ≠ ∅ , then for all a * ∈ E ( S ) ∩ L * a , we have the following two cases:

(1) if a ∈ T, then a * ∈ L * a ⊆ T since T is a left *–ideal of S, and so C _T  (a^*) =1 = C _T  (a) ;

(2) if a ∉ T, then C _T  (a^*) ≥0 = C _T  (a) .

Thus C _T is a fuzzy left *–ideal of S . Dually, C _T is a fuzzy right *–ideal of S for any right *–ideal T of S .□

Proposition 3.3. Let S be an arbitrary semigroup and a ∈ S. Define two fuzzy subsets f _R and g _L of S as follows: f R ( x ) = { t if x ∈ R * ( a ) 0 otherwise , where 0 < t < 1 ;(2) g L ( x ) = { t if x ∈ L * ( a ) 0 otherwise , where 0 < t < 1 .(3) Then f _R [resp., g _L ] is a fuzzy right [resp., left]*–ideal of S .

Proof. Let x, y ∈ S . If x ∈ R * ( a ) , then xy ∈ R * ( a ) since R * ( a ) is a right *–ideal of S . Hence f _R  (xy) = t = f _R  (x) ; if x ∉ R * ( a ) , then f _R  (xy) ≥0 = f _R  (x) . Therefore, f _R is a fuzzy right ideal of S .

Now, we show that f _R is a fuzzy right *–ideal of S . If x ∈ S and R * x ∩ E ( S ) ≠ ∅ , then for all x + ∈ R * x ∩ E ( S ) , there exist two cases:

(1) if x ∉ R * ( a ) , then f _R  (x⁺)≥0 = f _R  (x) ;

(2) if x ∈ R * ( a ) , then we can prove that x R * a , and so x + R * a . Hence x + ∈ R * a ⊆ R * ( a ) since R * ( a ) is the smallest right *–ideal of S containing a . Thus f _R  (x⁺) = t = f _R  (x) .

Summarizing the above arguments, we conclude that f _R is a fuzzy right *–ideal of S . Dually, g _L is a fuzzy left *–ideal of S .□

Theorem 3.4. Let S be a semigroup and a ∈ S. Define a fuzzy subset of S as follows: f I ( x ) = { t if x ∈ J * ( a ) 0 otherwise , where 0 < t < 1 .(4)

Then f _I is a fuzzy *–ideal of S .

Proof. An argument similar to that in the proof of Proposition 3.3 can show easily that f _I satisfies the condition (i) of Definition 3.1. Now, we only prove that f _I satisfies the condition (ii) of Definition 3.1. To see it, let x ∈ S and L * x ∩ E ( S ) ≠ ∅ . Then for all x * ∈ L * x ∩ E ( S ) , there are the following two facts:

(1) if x ∉ J * ( a ) , then f _I  (x^*)≥0 = f _I  (x) ;

(2) if x ∈ J * ( a ) , then x * ∈ L * x ⊆ J * ( a ) since J * ( a ) is a *–ideal of S . Hence f _I  (x^*) = t = f _I  (x) .

Dually, we can show that for all x + ∈ R * x ∩ E ( S ) , f _I  (x⁺) ≥0 = f _I  (x) . Therefore, f _I satisfies the condition (ii) of Definition 3.1. This completes the proof.

□

Proposition 3.5. Let f₁ be a fuzzy right *–ideal of an arbitrary semigroup S with idempotents and e ∈ E (S) such that f₁ (e) >0 . Define a fuzzy subset f _r of S as follows: f r ( x ) = { f 1 ( x ) if x ∈ eS 0 otherwise(5)

Then f _r is a fuzzy right *–ideal of S .

Proof. We first show that f _r is a fuzzy right ideal of S . To see it, let x, y ∈ S . If x ∈ eS, then xy ∈ eS . Hence f _r  (xy) = f₁ (xy)≥ f₁ (x) = f _r  (x) ; if x ∉ eS, then f _r  (xy) ≥0 = f _r  (x) . Therefore, f _r is a fuzzy right ideal of S .

Now, we prove that f _r is a fuzzy right *–ideal of S . Let x ∈ S and R * x ∩ E ( S ) ≠ ∅ . For all x + ∈ R * x ∩ E ( S ) , we consider the following two cases:

(1) if x ∉ eS, then f _r  (x⁺)≥0 = f _r  (x) ;

(2) if x ∈ eS, then x + ∈ R * x ⊆ eS since eS is a right *–ideal of S . Hence f _r  (x⁺) = f₁ (x⁺) ≥ f₁ (x) = f _r  (x) since f₁ is a fuzzy right *–ideal of S .

Summarizing the above cases, it can be immediately seen that f _r is a fuzzy right *–ideal of S .□

Proposition 3.6. Let f₁ be a fuzzy right *–ideal of an arbitrary semigroup S and e, h ∈ E (S) such that f₁ (e) >0, f₁ (h) >0 . Define a fuzzy subset f ^r of S as follows: f r ( x ) = { f 1 ( x ) if x ∈ eS ∪ hS 0 otherwise(6)

Then f ^r is a fuzzy right *–ideal of S .

Proof. It is a same proof of Proposition 3.5.□

Proposition 3.7. Suppose that f₁, f₂ are two fuzzy right *–ideals of an arbitrary semigroup S and e, h ∈ E (S) satisfying f₁ (e) >0, f₂ (h) >0 and eS ∩ hS = ∅ . Define a fuzzy subset f _rr of S as follows: f rr ( x ) = { f 1 ( x ) if x ∈ eS f 2 ( x ) if x ∈ hS 0 if x ∈ S \ ( eS ∪ hS )(7)

Then f _rr is a fuzzy right *–ideal of S .

Proof. Let x, y ∈ S . If x ∈ eS, then xy ∈ eS . Hence, f _rr  (xy) = f₁ (xy) ≥ f₁ (x) = f _rr  (x) since f₁ is a fuzzy right *–ideal of S; if x ∈ hS, then xy ∈ hS . Hence, f _rr  (xy) = f₂ (xy) ≥ f₂ (x) = f _rr  (x) since f₂ is a fuzzy right *–ideal of S ; if x ∉ eS and x ∉ hS, then f _rr  (xy) ≥0 = f _rr  (x) . Thus f _rr is a fuzzy right ideal of S .

Now, we prove that f _rr is a fuzzy right *–ideal of S . Suppose x ∈ S and R * x ∩ E ( S ) ≠ ∅ . For all x + ∈ R * x ∩ E ( S ) , consider the following three cases:

(1) if x ∈ eS, then x + ∈ R * x ⊆ eS since eS is a right *–ideal of S . Hence f _rr  (x⁺) = f₁ (x⁺) ≥ f₁ (x) = f _rr  (x) since f₁ is a fuzzy right *–ideal of S ;

(2) if x ∈ hS, then x + ∈ R * x ⊆ hS since hS is a right *–ideal of S . Hence f _rr  (x⁺) = f₂ (x⁺) ≥ f₂ (x) = f _rr  (x) since f₂ is a fuzzy right *–ideal of S ;

(3) if x ∉ eS and x ∉ hS, then f _rr  (x⁺) ≥0 = f _rr  (x) .

Therefore, f _rr is a fuzzy right *–ideal of S . This completes the proof.□ Next, we consider a construction of a fuzzy left (right) *–ideal of an abundant semigroup.

Theorem 3.8. Let T be a subset of an abundant semigroup S. Then T is a left [resp., right] *–ideal of S if and only if C _T is a fuzzy left [resp., right] *–ideal of S .

Proof. Evidently, T is a left [resp., right] ideal of S if and only if C _T is a fuzzy left [resp., right] ideal of S . By Theorem 3.2, the necessity is clear. We only show the sufficiency. Let C _T be a fuzzy left *–ideal of S and a ∈ T . Then L * a ⊆ T . In fact, let x ∈ L * a . Then there is a * ∈ L * a ∩ E ( S ) such that x L * a L * a * since S is abundant. Hence, by Lemma 2.1, C _T  (x) = C _T  (xa^*) ≥ C _T  (a^*) ≥ C _T  (a) =1, that is, C _T  (x) =1, and so x ∈ T . This gives that L * a ⊆ T . Therefore, T is a left *–ideal of S. Dually, we can show that T is a right *–ideal of S if and only if C _T is a fuzzy right *–ideal of S .□

Corollary 3.9. Let S be an abundant semigroup S. Then the following statements hold:

(1) for all a ∈ S, there is e ∈ E (S) such that f _R  = f _r  ;

(2) I is a *–ideal of S if and only if C _I is a fuzzy *–ideal of S .

Proof. (1) It follows from Theorem 3.8, Propositions 3.3 and 3.5.

(2) It follows from Theorem 3.8.□

In this section, we consider properties of fuzzy left [resp. right] *–ideals and fuzzy *–ideals of abundant semigroups, and give sufficient and necessary conditions for an arbitrary semigroup to be abundant.

Proposition 4.1. Let S be an abundant semigroup. Then for all fuzzy left [resp., right] *–ideal f of S, the following statements hold:

(1) for all a ∈ S, f (a) = f (a^*) [resp., f (a⁺) = f (a)], where a * ∈ L * a ∩ E ( S ) and a + ∈ R * a ∩ E ( S ) ;

(2) for all a, b ∈ S,   a L * b or a R * b implies f (a) = f (b) .

Proof. (1) Let f be a fuzzy left *–ideal of S and a ∈ S . Then L * a ∩ E ( S ) ≠ ∅ since S is abundant. Hence, for all a * ∈ L * a ∩ E ( S ) , a = aa^* from Lemma 2.1. Thus f (a) = f (aa^*) ≥ f (a^*), this together with f (a^*) ≥ f (a) implies that f (a) = f (a^*). Similarly, f (a⁺) = f (a) for all a ∈ S if f is a fuzzy right *–ideal of S.

(2) Suppose that f is a fuzzy left *–ideal of S. Let a, b ∈ S such that a L * b . Then there are a * ∈ L * a ∩ E ( S ) and b * ∈ L * b ∩ E ( S ) such that a * L * a L * b L * b * since S is abundant. Hence, by Lemma 2.1, f (a) = f (ab^*) ≥ f (b^*) ≥ f (b) = f (ba^*) ≥ f (a^*) ≥ f (a) since f is a fuzzy left *–ideal of S. Therefore, f (a) = f (b) . Dually, a R * b implies f (a) = f (b) for all fuzzy right *–ideal f of S.□

Theorem 4.2. Let f and g be a fuzzy right *–ideal and a fuzzy left *–ideal of an abundant semigroup S, respectively. Then the following statements hold:

(1) f ∘ g = f ∩ g;

(2) f ∘ f = f and g ∘ g = g .

Proof. (1) Let a ∈ S . Then L * a ∩ E ( S ) ≠ ∅ and R * a ∩ E ( S ) ≠ ∅ since S is abundant. Hence

f ∘ g ( a ) = ∨ a = xy { f ( x ) ∧ g ( y ) } ≥ { f ( a + ) ∧ g ( a ) } ∨ { f ( a ) ∧ g ( a * ) } ≥ { f ( a ) ∧ g ( a ) } ∨ { f ( a ) ∧ g ( a ) } = f ( a ) ∧ g ( a ) = ( f ∩ g ) ( a ) , where a * ∈ L * a ∩ E ( S ) and a + ∈ R * a ∩ E ( S ) . Thus f ∘ g ⊇ f ∩ g . On the other hand, f ∘ g ⊆ f ∘ S ⊆ f and f ∘ g ⊆ S ∘ g ⊆ g, that is, f ∘ g ⊆ f ∩ g . Therefore, (1) holds.

(2) Let a ∈ S . Then, by hypothesis, R * a ∩ E ( S ) ≠ ∅ . Hence f ∘ f (a) = ∨ _a=xy {f (x) ∧ f (y)} ≥ f (a⁺) ∧ f (a) ≥ f (a) ∧ f (a) = f (a) , where a + ∈ R * a ∩ E ( S ) . Therefore, f ∘ f ⊇ f . On the other hand, f ∘ f ⊆ f ∘ S ⊆ f since f is a fuzzy right *–ideal of S . Therefore, f ∘ f = f . Similarly, g ∘ g = g .□ As an immediate result of Theorem 4.2, we have

Theorem 4.3. Every fuzzy *–ideal of an abundant semigroup is idempotent.

Remark 4.1 (1) The condition “ fuzzy *–ideal " in Theorem 4.3 can not be weakened to “ fuzzy ideal ". In fact, in our Example 3.2, we have proved that S is an abundant semigroup. If we define a fuzzy subset of S as follows: f : S ⟶ [ 0 , 1 ] , f ( a ) = 1 - a We can easily check that f is a fuzzy ideal of S . However, for all a ≠ 1, it is easy to see that (f ∘ f) (a) = {f (a) ∧ f (1)} =0 ≠ f (a) , that is, f ∘ f ≠ f. Therefore, it is not true that every fuzzy ideal of an abundant semigroup is idempotent.

(2) By Theorem 4.3, every fuzzy ideal of a regular semigroup is idempotent since an arbitrary regular semigroup is abundant and its fuzzy ideals are fuzzy *–ideals.

Example 4.1. Let S = { [ x ] 2 × 2 | x ∈ ℕ } ⋃ { [ 1 2 ] 2 × 2 } , where ℕ is the set of all non-negative integers and [x] _2×2 denotes the following second-order matrix ( x x x x ) for all x ∈ ℕ ∪ { 1 2 } . Then it is seen easily that S is an abundant semigroup with respect to the general matrix multiplication, and that E ( S ) = { [ 0 ] 2 × 2 , [ 1 2 ] 2 × 2 } . In fact, it is routine to check that L * classes and R * classes of S are both {[0] _2×2} and S \ {[0] _2×2}. In other words, for all [a] _2×2 ∈ S, we have that [ a ] 2 × 2 * = [ a ] 2 × 2 + = [ 0 ] 2 × 2 or [ a ] 2 × 2 * = [ a ] 2 × 2 + = [ 1 2 ] 2 × 2 .

Consider a fuzzy subset of S as follows: f : S ⟶ [ 0 , 1 ] ] , f ( [ x ] 2 × 2 ) = { 1 if x = 0 1 2 otherwise(8) Then f ( [ x ] 2 × 2 · [ y ] 2 × 2 ) = f ( [ 2 xy ] 2 × 2 ) = f ( [ x ] 2 × 2 ) = f ( [ y ] 2 × 2 ) = 1 2 if xy ≠ 0 . Otherwise, f ([x] _2×2 · [y] _2×2) = f ([2xy] _2×2) =1 ≥ f ([x] _2×2) and f ([x] _2×2 · [y] _2×2) ≥ f ([y] _2×2)). This means that f satisfies the condition (i) of Definition 3.1. On the other hand, we have f ( [ x ] 2 × 2 * ) = f ( [ x ] 2 × 2 + ) = f ( [ 0 ] 2 × 2 ) = 1 ≥ f ( [ x ] 2 × 2 ) or f ( [ x ] 2 × 2 * ) = f ( [ x ] 2 × 2 + ) = f ( [ 1 2 ] 2 × 2 ) = 1 2 = f ( [ x ] 2 × 2 ) . Therefore, f satisfies the condition (ii) of Definition 3.1. That is, f is a fuzzy *–ideal of S .

Finally, we show that f ∘ f = f . To see it, let [x] _2×2 ∈ S . If x = 0, then f ∘ f = f is clear. If x ≠ 0, then there are [y] _2×2, [z] _2×2 ∈ S \ {[0] _2×2} such that [x] _2×2 = [y] _2×2 · [z] _2×2. Hence ( f ∘ f ) ( [ x ] 2 × 2 ) = ⋁ [ x ] 2 × 2 = [ y ] 2 × 2 · [ z ] 2 × 2 { f ( [ y ] 2 × 2 ) ∧ f ( [ z ] 2 × 2 ) } = ⋁ [ x ] 2 × 2 = [ y ] 2 × 2 · [ z ] 2 × 2 { 1 2 ∧ 1 2 } = 1 2 = f ( [ x ] 2 × 2 ) . Therefore, f ∘ f = f .

Theorem 4.4. Let S be an abundant semigroup and a, b ∈ S. Put A = a⁺S,  B = b⁺S, where a + ∈ R * a ∩ E ( S ) and b + ∈ R * b ∩ E ( S ) . Then the following statements are equivalent:

(1) a R * b ;

(2) C _A  (b) = C _B  (a) =1 ;

(3) C _I  (a) = C _I  (b) for all right *–ideals I of S ;

(4) f (a) = f (b) for all fuzzy right *–ideals f of S .

Proof. By Theorem 3.8 and Proposition 4.1(2), (4)⇒ (3) ⇒ (2) and (1) ⇒ (4) are clear. We only need to show that (2) ⇒ (1).

Let a, b ∈ S such that C _A  (b) = C _B  (a) =1, where A = a⁺S,  B = b⁺S, a + ∈ R * a ∩ E ( S ) and b + ∈ R * b ∩ E ( S ) . Then a ∈ b⁺S and b ∈ a⁺S. Hence, a + ∈ R * a ⊆ b + S and b + ∈ R * b ⊆ a + S since a⁺S and b⁺S are right *–ideals of S. Thus a⁺ = b⁺a⁺ and b⁺ = a⁺b⁺, and so a + R b + . Therefore, a R * a + R b + R * b . That is, a R * b since R ⊆ R * . This completes the proof.□

We remark that the dual of Theorem 4.4 is true. Next, we shall consider the relationship between fuzzy *–ideals and abundance for an arbitrary semigroup with idempotents.

Theorem 4.5. Let S be an arbitrary semigroup. Then S is abundant if and only if for every a ∈ S, there exist e, h ∈ E (S) such that C _I  (a) = C _I  (e) and C _J  (a) = C _J  (h) for all left *–ideals I and all right *–ideals J of S .

Proof. Let S be abundant and a ∈ S. Then there are e, h ∈ E (S) such that a L * e and a R * h . Hence, by Theorem 4.4 and its dual, C _I  (a) = C _I  (e) and C _J  (a) = C _J  (h) for all left *–ideals I and all right *–ideals J of S .

Conversely, if for all a ∈ S, there are e, h ∈ E (S) such that C _I  (a) = C _I  (e) and C _J  (a) = C _J  (h) for all left *–ideals I and all right *–ideals J of S . Choose Se to replace I . We have a ∈ Se . Hence L * ( a ) ⊆ Se since L * ( a ) is the smallest left *–ideal of S containing a . Choose L * ( a ) to replace I . We have e ∈ L * ( a ) , and so Se ⊆ L * ( a ) since Se is the smallest left *–ideal of S containing e . Thus, L * ( a ) = Se . Similarly, R * ( a ) = hS . Therefore S is abundant.□

As an immediate corollary of Proposition 4.1(1), Theorems 4.4 and its dual, and 4.5, we have

Corollary 4.6. Let S be an arbitrary semigroup. Then S is abundant if and only if for every a ∈ S, there exist e, h ∈ E (S) such that f (a) = f (e) and g (a) = g (h) for all fuzzy left *–ideals f and all fuzzy right *–ideals g of S .

As an application of Corollary 4.6, Theorem 4.4 and its dual, the following corollaries are true.

Corollary 5.1. Let S be an arbitrary semigroup. Then S is regular if and only if for every a ∈ S, there exist e, h ∈ E (S) such that f (a) = f (e) and g (a) = g (h) for all fuzzy left ideals f and all fuzzy right ideals g of S .

Proof. The sufficiency is clear from Lemma 2.2, so we only show the necessity. Note that a regular semigroup is abundant, and that a fuzzy ideal of a regular semigroup is a fuzzy *–ideal. By Corollary 4.6, the necessity holds.□

(1) if a ∈ E (S), then it is evident that the conditions of Corollary 5.1 hold;

(2) if a ∈ S \ E (S) , then a = x or a = y. Hence, f (a) = f (x) = f (xyx) ≥ f (yx) = f (h) = f (yx) ≥ f (x) or f (a) = f (y) = f (yxy) ≥ f (xy) = f (e) = f (xy) ≥ f (y) since f is a fuzzy left ideal of S . That is, f (a) = f (h) or f (a) = f (e), where e, h ∈ E (S) . Similarly, g (a) = g (x) = g (xyx) ≥ g (xy) = g (e) = g (xy) ≥ g (x) or g (a) = g (y) = g (yxy) ≥ g (yx) = g (h) = g (yx) ≥ g (y) since g is a fuzzy right ideal of S . That is, g (a) = g (e) or g (a) = g (h), where e, h ∈ E (S) .

Summarizing the above two cases, it can be immediately seen that the conditions of Corollary 5.1 hold.

Note that a regular semigroup S is inverse if each L class and each R class of S contains a unique idempotent. As an immediate corollary of Corollary 5.1, Theorem 4.4 and its dual, we have

Corollary 5.2. Let S be an arbitrary semigroup. Then S is an inverse semigroup if and only if for every a ∈ S, there exists a unique element e ∈ E (S) such that f (a) = f (e) and there exists a unique element h ∈ E (S) such that g (a) = g (h) for all fuzzy left ideals f and all fuzzy right ideals g of S .

Example 5.2. Let S = { [ x ] 2 × 2 | x ∈ ℕ } ⋃ { [ 1 x ] 2 × 2 | x ∈ ℕ \ { 0 } } , where ℕ is the set of all non-negative integers and [x] _2×2 denotes the following matrix ( x x x x ) . Then it can be seen easily that S is an inverse semigroup with respect to the general matrix multiplication, and that E ( S ) = { [ 0 ] 2 × 2 , [ 1 2 ] 2 × 2 } . In fact, it is routine to check that [ 1 4 x ] 2 × 2 is the inverse of [x] _2×2 for all x ≠ 0. Suppose that f and g are a fuzzy left ideal and a fuzzy right ideal of S, respectively. Consider the following two facts:

(1) if a = 0, then f ([a] _2×2) = f ([0] _2×2) and g ([a] _2×2) = g ([0] _2×2), where [0] _2×2∈ E (S) ;

(2) if a ≠ 0, then f ( [ a ] 2 × 2 ) = f ( [ a ] 2 × 2 [ 1 4 a ] 2 × 2 [ a ] 2 × 2 ) ≥ f ( [ 1 4 a ] 2 × 2 [ a ] 2 × 2 ) = f ( [ 1 2 ] 2 × 2 ) = f ( [ 1 4 a ] 2 × 2 [ a ] 2 × 2 ) ≥ f ( [ a ] 2 × 2 ) , which implies f ( [ a ] 2 × 2 ) = f ( [ 1 2 ] 2 × 2 ) , where [ 1 2 ] 2 × 2 ∈ E ( S ) . Dually, g ( [ a ] 2 × 2 ) = g ( [ a ] 2 × 2 [ 1 4 a ] 2 × 2 [ a ] 2 × 2 ) ≥ g ( [ a ] 2 × 2 [ 1 4 a ] 2 × 2 ) = g ( [ 1 2 ] 2 × 2 ) = g ( [ a ] 2 × 2 [ 1 4 a ] 2 × 2 ) ≥ g ( [ a ] 2 × 2 ) . That is, g ( [ a ] 2 × 2 ) = g ( [ 1 2 ] 2 × 2 ) , where [ 1 2 ] 2 × 2 ∈ E ( S ) .

Therefore, we have that the conditions of Corollary 5.2 hold.

6 Conclusions

As we know, fuzzy ideals of semigroups play an important role in the characterization of semigroups structures. The fuzzy Green’s relations [8] are useful in studying the structure, and in particular, the decompositions of (ordered) semigroups. The purpose of this paper is to establish the relationships between fuzzy ideals satisfying some conditions and abundance for an arbitrary semigroup. To see it, the paper introduces the notion of a fuzzy *–ideal of a semigroup. The relationships between fuzzy *–ideals and abundance for an arbitrary semigroup are obtained by exploiting Green *–relations. As an application, some new sufficient and necessary conditions for an arbitrary semigroup to be a regular (an inverse) semigroup are obtained by using fuzzy *–ideals of semigroups. In fact, results given in this paper can be applied to the theory of regular semigroups, and might be useful for further study of fuzzy algebra theory.