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Open Mathematics

formerly Central European Journal of Mathematics

Editor-in-Chief: Gianazza, Ugo / Vespri, Vincenzo


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Volume 16, Issue 1

Issues

Volume 13 (2015)

On 𝓠-regular semigroups

Xinyang Feng
Published Online: 2018-05-30 | DOI: https://doi.org/10.1515/math-2018-0048

Abstract

In this paper, we give some characterizations of 𝓠-regular semigroups and show that the class of 𝓠-regular semigroups is closed under the direct product and homomorphic images. Furthermore, we characterize the 𝓠-subdirect products of this class of semigroups and study the E-unitary 𝓠-regular covers for 𝓠-regular semigroups, in particular for those whose maximum group homomorphic image is a given group. As an application of these results, we claim that the similar results on V-regular semigroups also hold.

Keywords: 𝓠-regular semigroup; 𝓠-subdirect product; Surjective 𝓠-subhomomorphism; E-unitary 𝓠-regular cover

MSC 2010: 20M10

1 Introduction and Preliminaries

Let S be a regular semigroup. Let V(a) be the set of all inverses of a for each aS. We also use E(S) to denote the set of all idempotents in S. It is well known that a regular semigroup S is orthodox if and only if for all a, bS,

V(b)V(a)V(ab).

The concept of 𝓟-regular semigroups was first introduced by Yamada and Sen [1]. In [2], Zhang and He characterized the structure of 𝓟-regular semigroups. In fact, the class of 𝓟-regular semigroups is a generalization of orthodox semigroups and regular *-semigroups.

On the other hand, by Onstad [3], a regular semigroup is said to be a V-regular semigroup if for all a, bS,

V(ab)V(b)V(a).

According to the definition of orthodox semigroups, V-regular semigroup is a dual form of orthodox semigorup. Evidently, a regular semigroup S is both orthodox and V-regular if and only if for all a, bS,

V(ab)=V(b)V(a).

The class of inverse semigroups forms the most important class of regular semigroups which satisfy the above condition. As a generalization of inverse semigroups and orthodox semigroups, Gu and Tang [4] investigated Vn-semigroups and showed that the class of Vn-semigroups is closed under direct products and homomorphic images.

In V-regular semigroups case, Nambooripad and Pastijn [5] gave a characterization of this class of semigroups and Zheng and Ren [6] described congruences on V-regular semigroups in terms of certain congruence pairs. Then, Li [7] generalized the concept of V-regular semigroups and investigated 𝓠-regular semigroups which is a dual form of 𝓟-regular semigroups.

A regular semigroup S is called 𝓠-regular if for any aS there exists a non-empty set VQV(a) satisfying the following conditions:

  1. aa+VQ(aa+) and a+aVQ(a+a) for any aS and a+VQ(a),

  2. VQ(ab) ⊆ VQ(b)VQ(a) for all a, bS, where a+ satisfying the above conditions is called a 𝓠-inverse of a and VQ(a) denotes the set of all 𝓠-inverses of a.

In [7], Li also gave an example of 𝓠-regular semigroup and showed that this class of semigroups properly contains the class of V-regular semigroups. At the same time, an equivalent characterization of 𝓠-regular semigroups was obtained.

A regular semigroup S is 𝓠-regular if and only if there exists a set QE(S) satisfying the following conditions:

  1. QLa ≠ ∅ and QRa ≠ ∅ for any aS;

  2. ωl|Q = ω|Q ∘ 𝓛|Q and ωr|Q = ω|Q ∘ 𝓡|Q;

  3. For any a, bS, if (ab)+V(ab) such that (ab)+(ab), (ab)(ab)+Q, then there exists e1QLa and f2QRa such that b(ab)+a = f2e1.

In this paper, a 𝓠-regular semigroup S will be denoted by S(Q). A subset Q of E(S) satisfying (1)-(3) is called a charcateristic set (for simple a C-set) of S.

Some results on subdirect products of inverse semigroups were characterized by McAlister and Reilly [8]. In [9], Nambooripad and Veeramony discussed the subdirect products of regular semigroups. Mitsch [10] studied the subdirect products of E-inversive semigroups. In [11], Zheng characterized the subdirect products of 𝓟-regular semigroups. Throughout this paper, we investigate some properties of 𝓠-regular semigroups at first and characterize the 𝓠-subdirect products of this class of semigroups. Furthermore, we introduce the concept of the E-unitary 𝓠-regular covers for 𝓠-regular semigroups, in particular for those whose maximum group homomorphic image is a given group. Finally, we deduce the similar results on V-regular semigroups also hold up.

For notations and definitions given in this paper, the reader is referred to Howie [12].

2 On 𝓠-regular semigroups

As a dual form of 𝓟-regular semigroups, Li [7] introduced the concept of 𝓠-regular semigroups. In this section, we give some characterizations of 𝓠-regular semigroups. In particular, we show that the class of 𝓠-regular semigroups is closed under the homomorphic images.

Proposition 2.1

Let S(Q) be a 𝓠-regular semigroup. For any pQ,

VQ(p)=(QLp)(QRp).

Proof

Since pQE(S), QLp ≠ ∅, QRp ≠ ∅. For any hVQ(p), we have h = hph = (hp)(ph) ∈ (QLp)(QRp), and so VQ(p) ⊆ (QLp)(QRp).

Conversely, let fQLp, gQRp. Then

p(fg)p=(pf)(gp)=pp=p,(fg)p(fg)=f(gp)fg=fpfg=fg.

So fgV(p). Additionally,

(fg)p=f(gp)=fp=fQ,p(fg)=(pf)g=pg=gQ.

Thus fgVQ(p), namely, (QLp)(QRp) ⊆ VQ(p). Therefore, VQ(p) = (QLp)(QRp). □

Proposition 2.2

([13]). Let S(Q) be a 𝓠-regular semigroup. If aS(Q), a+VQ(a), then

VQ(a)=VQ(a+a)a+VQ(aa+).

Theorem 2.3

Let S(Q) be a 𝓠-regular semigroup. If eQLa, fQRa, for any aS, then there exists a unique a+VQ(a) such that a+a = e, aa+ = f.

Proof

For any aS, let eQLa and fQRa. And so there is an inverse a′ of a in ReLf such that aa = eQ, aa′ = fQ. Thus a′ ∈ VQ(a). If there exists a a+VQ(a) such that a+a = e, aa+ = f, then

a=aaa=af=aaa+=ea+=a+aa+=a+. □

Corollary 2.4

Let S(Q) be a 𝓠-regular semigroup. For any a, bS,

  1. (a, b) ∈ 𝓛 if and only if there exists a+VQ(a), b+VQ(b) such that a+a = b+b;

  2. (a, b) ∈ 𝓡 if and only if there exists a+VQ(a), b+VQ(b) such that aa+ = bb+;

  3. (a, b) ∈ 𝓗 if and only if there exists a+VQ(a), b+VQ(b) such that aa+ = bb+, a+a = b+b.

Proof

(1) For any a+VQ(a), we have a+aQLa. If (a, b) ∈ 𝓛 and there exists a eQRb, then e 𝓡 b 𝓛 a 𝓛 a+a. By Theorem 2.3, there exists a unique b+VQ(b) such that b+b = a+a.

On the other hand, if there exist a+VQ(a), b+VQ(b) such that b+b = a+a, then a 𝓛 a+a = b+b 𝓛 b, that is a 𝓛 b.

By using similar arguments as the above, we can proof (2) and (3). □

Corollary 2.5

Let S(Q) be a 𝓠-regular semigroup. If e, fQ, then (e, f) ∈ 𝓓 if and only if there exist aS, a+VQ(a) such that a+a = f, aa+ = e.

Proof

If (e, f) ∈ 𝓓, there exists aReLf. By Theorem 2.3, there exists a+VQ(a) such that a+a = f, aa+ = e.

Conversely, if there exist aS, a+VQ(a) such that a+a = f, aa+ = e, then e 𝓡 a, a 𝓛 f. Hence e 𝓓 f. □

Proposition 2.6

Let S(Q) be 𝓠-regular semigroup. If q 𝓡 p (q 𝓛 p) for any p, qQ, then qVQ(p).

Proof

Let qRp. Since p, qQE(S), so pq = q, qp = p. Hence pqp = qp = p, qpq = pq = q, namely qV(p). For another, pq = qQ, qp = pQ. Hence qVQ(p). In the case of p 𝓛 q, the proof is similar. □

Corollary 2.7

Let S(Q) be a 𝓠-regular semigroup. Then the following statements are equivalent:

  1. for any q, pQ, if VQ(q) ∩ VQ(p) ≠ ∅, then VQ(q) = VQ(p);

  2. for any e, fE(S), if VQ(e) ∩ VQ(f) ≠ ∅, then VQ(e) = VQ(f);

  3. for any a, bS(Q), if VQ(a) ∩ VQ(b) ≠ ∅, then VQ(a) = VQ(b).

Proof

We only need to proof (1) ⇒ (3). Let cVQ(a) ∩ VQ(b). By Proposition 2.2,

VQ(a)=VQ(ca)cVQ(ac),VQ(b)=VQ(cb)cVQ(bc).

Since ca, ac, cb, bcQ and ca 𝓡 c 𝓡 cb, ac 𝓛 c 𝓛 bc, by Proposition 2.6,

caVQ(ca)VQ(cb),acVQ(ac)VQ(bc).

By (1),

VQ(ca)=VQ(cb),VQ(ac)=VQ(bc).

Thus, VQ(a) = VQ(b). □

Proposition 2.8

Let S(Q) be a 𝓠-regular semigroup and ρ an idempotent-separating congruence on S. For any a, bS, if a ρ b, then a+ ρ b+ for some a+VQ(a), b+VQ(b).

Proof

Since ρ is an idempotent-separating congruence on S, a ρ b implies that a 𝓗 b. By Corollary 2.4, there exist a+VQ(a), b+VQ(b) such that aa+ = bb+, a+a = b+b. Thus

a+=a+aa+=a+bb+,b+=b+bb+=a+ab+.

Since ρ is a congruence, a+ab+ ρ a+bb+, that is b+ ρ a+. □

Theorem 2.9

Let S(Q) be a 𝓠-regular semigroup and T a regular semigroup. If ψ : S(Q) → T is a semigroup homomorphism and = { : qQ}, then T() is 𝓠-regular.

Proof

For any aS, there exists a a+VQ(a) such that aa+, a+aQ. And so

(aψ)(a+ψ)(aψ)=(aa+a)ψ=aψ,(a+ψ)(aψ)(a+ψ)=(a+aa+)ψ=a+ψ.

That is a+ψV(). Since (aψ)(a+ψ) = (aa+)ψ, a+ψV() ≠ ∅. It is easy to see that

(aψ)(a+ψ)VQ((aψ)(a+ψ))and(a+ψ)(aψ)VQ((a+ψ)(aψ)).

On the other hand, for any (ab)+VQ(ab), since S is 𝓠-regular, there exist a+VQ(a), b+VQ(b) such that

(ab)+ψ=(b+a+)ψ=(b+ψ)(a+ψ).

Thus V((ab)ψ) ⊆ V()V() and T() is 𝓠-regular. □

3 𝓠-subdirect products of 𝓠-regular semigroups

Some results on subdirect products of regular semigroups and E-inversive semigroups were characterized by Nambooripad [9] and Mitsch [10]. In this section, we show that the class of 𝓠-regular semigroups is closed under the direct product and characterize the 𝓠-subdirect products of 𝓠-regular semigroups.

For arbitrary semigroup S, Petrich [14] introduced the concept of filters of S, that is, a subsemigroup F of S such that abF implies aF and bF.

Example 3.1

Let S = {a, b, c, d, e} be the semigroup with operation defined by

Then F = {d, e} is a filter of S.

In what follows, we introduce the concept of 𝓠-inverse filters of a 𝓠-regular semigroup.

Definition 3.2

A regular subsemigroup T of 𝓠-regular semigroup S(Q) is called a 𝓠-inverse filter, if

  1. T(QET) is a 𝓠-regular semigroup;

  2. For any t1, t2T and t1+VQS(t1),t2+VQS(t2),ift1+t2+T,thent1+,t2+T.

Definition 3.3

Let S1(Q1) and S2(Q2) be two 𝓠-regular semigroups. A homomorphism f of S1(Q1) into S2(Q2) is called a 𝓠-homomorphism if Q1f = Q2S1(Q1)f. A 𝓠-homomorphism f : S1(Q1) → S2(Q2) is called a 𝓠-isomorphism if f is bijective, in such the case, S1(Q1) is called 𝓠-isomorphic to S2(Q2), and denoted by S1(Q1) Q S2(Q2).

Proposition 3.4

Let S1(Q1) and S2(Q2) be two 𝓠-regular semigroups. If S(Q) = S1(Q1) × S2(Q2), where Q = {(p1, p2): p1Q1, p2Q2}, then S(Q) is a 𝓠-regular semigroup.

Proof

Obviously, S(Q) is a semigroup. If (s, t) ∈ S, where sS1, tS2, since S1, S2 are all 𝓠-regular, there exists s′ ∈ VS1(s), t′ ∈ VS2(t) such that (s′, t′) ∈ S. It is easy to see that (s′, t′) ∈ V(s, t), and so S is regular. Let

VQ1(s)×VQ2(t)={(s+,t+):s+VQ1(s),t+VQ2(t)}.

Clearly, it is non-empty and contained in V(s, t). For any (s+, t+) ∈ VQ1(s) × VQ2(t), we have

(s,t)(s+,t+)VQ1(ss+)×VQ2(tt+),(s+,t+)(s,t)VQ1(s+s)×VQ2(t+t).

For another, let (s, t), (x, y) ∈ S. For any ((sx)+, (ty)+) ∈ VQ1(sx) × VQ2(ty), since S1, S2 are all 𝓠-regular, there exists s+VQ1(s), x+VQ1(x), t+VQ2(t), y+VQ2(y) such that

((sx)+,(ty)+)=(x+s+,y+t+)=(x+,y+)(s+,t+)(VQ1(x)×VQ2(y))(VQ1(s)×VQ2(t)).

Hence, S(Q) is a 𝓠-regular semigroup. □

Definition 3.5

Let S1(Q1), S2(Q2) be 𝓠-regular semigroups and S(Q) be the direct product of them, where Q = {(p1, p2): p1Q1, p2Q2}. If T(Q′) is a 𝓠-inverse filter of S(Q) and the projections

f1:T(Q)S1(Q1),(s1,s2)s1,f2:T(Q)S2(Q2),(s1,s2)s2

are all surjective 𝓠-homomorphisms, then T(Q′) is called a 𝓠-subdirect product of S1(Q1) and S2(Q2).

Definition 3.6

Let S(Q1) and T(Q2) be two 𝓠-regular semigroups. A mapping φ : S → 2T (the power set of T) is called a surjective 𝓠-subhomomorphism of S(Q1) onto T(Q2), if it satisfies the following

  1. for any sS, sφ ≠ ∅;

  2. for any s1, s2S, (s1φ)(s2φ) ⊆ (s1s2)φ;

  3. sS sφ = T;

  4. for any tsφ (sS, tT), there exist s+VQ1(s), t+VQ2(t) such that t+s+φ;

  5. for any p1Q1, there exists p2Q2 such that p2p1φ; and for any p2Q2, there exists p1Q1 such that p2p1φ;

  6. for any t1s1φ, t2s2φ, if t1+t2+(s1+s2+)φ,thent1+s1+φ,t2+s2+φ,whereti+VQ2(ti),si+VQ1(si), i = 1, 2.

Theorem 3.7

Let S(Q1) and T(Q2) be 𝓠-regular semigroups, φ a surjective 𝓠-subhomomorphism of S(Q1) onto T(Q2). If

π=π(S,T,φ)={(s,t)S×T:tsφ},Q={(p1,p2)Q1×Q2:p2p1φ},

then π(Q) is a 𝓠-subdirect product of S(Q1) and T(Q2).

Conversely, every 𝓠-subdirect product of S(Q1) and T(Q2) can be constructed in this way.

Proof

If (s1, t1), (s2, t2) ∈ π, then t1s1φ, t2s2φ. By (2), t1t2 ∈ (s1s2)φ, and so (s1s2, t1t2) ∈ π. Hence π is a subsemigroup of S × T. If (s, t) ∈ π, then tsφ. By (4), there exist s+VQ1(s) and t+VQ2(t) such that t+s+φ, and so (s+, t+) ∈ π. It is easy to see that (s+, t+) ∈ V(s, t). Thus π is a regular semigroup. Let

VQ(s,t)={(s+,t+)VQ1(s)×VQ2(t):t+s+φ}.

If (s, t) ∈ π, then tsφ. By (4), VQ(s, t) ≠ ∅. For any (s, t) ∈ π, (s+, t+) ∈ VQ(s, t),

(s,t)(s+,t+)=(ss+,tt+),

where s+VQ1(s), t+VQ2(t), t+s+φ. Since S(Q1), T(Q2) are 𝓠-regular, it follows that ss+VQ1(ss+), tt+VQ2(tt+). And tt+ ∈ (ss+)φ, hence

(s,t)(s+,t+)VQ(ss+,tt+)=VQ((s,t)(s+,t+)).

Similarly, (s+, t+)(s, t) ∈ VQ((s+, t+)(s, t)).

On the other hand, let (s1, t1), (s2, t2) ∈ π. For any

((s1s2)+,(t1t2)+)VQ(s1s2,t1t2),

since S(Q1) and T(Q2) are all 𝓠-regular semigroups, there exist s1+VQ1(s1), s2+VQ1(s2), t1+VQ2(t1), t2+VQ2(t2) such that

((s1s2)+,(t1t2)+)=(s2+s1+,t2+t1+)=(s2+,t2+)(s1+,t1+).

Since t2+t1+(s2+s1+)φ, by (6), t2+s2+φ,t1+s1+φ. And so

(s2+,t2+)VQ(s2,t2),(s1+,t1+)VQ(s1,t1).

Hence,

VQ(s1s2,t1t2)VQ(s2,t2)VQ(s1,t1).

that is, π(Q) is a 𝓠-regular semigroup.

For any

(s1+,t1+)VQ1(s1)×VQ2(t1),(s2+,t2+)VQ1(s2)×VQ2(t2).

If (s1+,t1+)(s2+,t2+)π, namely (s1+s2+,t1+t2+)π, then t1+t2+(s1+s2+)φ. By (6), t1+s1+φ,t2+s2+φ, and so (s1+,t1+),(s2+,t2+)π. Hence π(Q) is a 𝓠-inverse filter of S(Q1) × T(Q2).

By (1) and (5), the projection f1: π → S, (s, t) ↦ s is a surjective 𝓠-homomorphism. By (3) and (5), the projection f2: π → T, (s, t) ↦ t is a surjection 𝓠-homomorphism. Therefore, π(Q) is a 𝓠-subdirect product of S(Q1) and T(Q2).

Conversely, let H(Q) be a 𝓠-subdirect product of S(Q1) and T(Q2). Let

φ:S2T,s{tT:(s,t)H}.

Since H(Q) is a 𝓠-subdirect product of S(Q1) and T(Q2), there exists tT such that (s, t) ∈ H for any sS. Thus tsφ. Hence (1) holds. Since H is a subsemigroup of S(Q1) × T(Q2), (2) holds. Since the projection f2: HT is surjective, (3) holds. If tsφ (sS, tT), then (s, t) ∈ H. Since H(Q) is 𝓠-regular, there exists

(s+,t+)(VQ1(s)×VQ2(t))H.

Hence s+VQ1(s), t+VQ2(t), and t+s+φ, so (4) holds. For any p1Q1, there exists (p1, p2) ∈ Q = H ∩ (Q1 × Q2), and so p2Q2 and p2p1φ. Additionally, since f2 is a surjective 𝓠-homomorphism, there exists

(p1,p2)Q=H(Q1×Q2)

for any p2Q2. Hence p1Q1 and p2p1φ. Thus (5) holds. Since H(Q) is a 𝓠-inverse filter of S(Q1) × T(Q2), (6) holds. By the proof above, φ is a surjective 𝓠-subhomomorphism of S(Q1) onto T(Q2). Obviously, H = π(S, T, φ). □

4 E-unitary covers for 𝓠-regular semigroups

McAlister and Reilly [8] have given E-unitary covers of inverse semigroups by using surjective subhomomorphisms of inverse semigroups. Mitsch [10] has given some sufficient conditions on E-unitary, E-inverse covers of E-inversive semigroups.

In this section, we introduce the concept of the E-unitary 𝓠-regular covers of 𝓠-regular semigroups, in particular for those whose maximum group homomorphic image is a given group.

Let S be a regular semigroup. A subset H of S is said to be

  1. full if E(S) ⊆ H;

  2. self-conjugate if aHa′ ⊆ H and aHaH for all aS and all a′ ∈ V(a).

Let U be the minimum full and self-conjugate subsemigroup of S.

Lemma 4.1

([15]). Let S be a regular semiroup. The minimum group congruence σ on S is given by

σ={(a,b)S×S:(x,yU)xa=by}.

In [12], a subset A of a semigroup S is called right unitary, if

(aA)(sS)saAsA.

A is called left unitary, if

(aA)(sS)asAsA.

A right and left unitary subset is called unitary.

A regular semigroup S is called E-unitary, if the set E(S) of idempotents of S is unitary.

Definition 4.2

Let T(Q1) and S(Q) be two 𝓠-regular semigroups. T(Q1) is called an E-unitary 𝓠-regular cover of S(Q), if T(Q1) is E-unitary, and there exists an idempotent separating 𝓠-homomorphism of T(Q1) onto S(Q).

A group G is a 𝓠-regular semigroup whose C-set is {1}, where 1 is the identity element of G.

Definition 4.3

Let S(Q) be a 𝓠-regular semigroup and G be a group. A surjective 𝓠-subhomomorphism φ of S(Q) onto G is called unitary, if

(sS)1sφsE(S),

where 1 is the identity element of G.

Theorem 4.4

Let S(Q) be a 𝓠-regular semigroup and G be a group. If there exists a unitary surjetive 𝓠-subhomomorphism φ of S(Q) onto G, then π (S, G, φ) is an E-unitary 𝓠-regular cover of S(Q).

Proof

By Theorem 3.7, T(Q1) = π(S, G, φ) is 𝓠-regular whose C-set is

Q1={(p,1)S×G:pQ,1pφ}.

For any (s, g) ∈ T, (e, 1) ∈ ET, if

(s,g)(e,1)ET{(t,1)S×G:tES},

then (se, g) ∈ ET, and so g = 1, and (s, 1) ∈ T(Q1). By the definition of T(Q1), 1 ∈ sφ. Since φ is unitary, sES. Hence (s, g) ∈ ET, so T(Q1) is E-unitary.

Define f : T(Q1) → S(Q), (s, g) ↦ s. Then f is a surjective 𝓠-homomorphism. Obviously, f is idempotent separating. Hence π(S, G, φ) is an E-unitary 𝓠-regular cover of S(Q). □

Definition 4.5

A 𝓠-regular semigroup T(Q1) is called an E-unitary 𝓠-regular cover of S(Q) through G, if

  1. T(Q1) is an E-unitary 𝓠-regular cover of S(Q);

  2. T/σ Q G, where σ is the minimum group congruence on T(Q1).

Lemma 4.6

Let S(Q1), T(Q2) be two 𝓠-regular semigroups and f : S(Q1) → T(Q2) be a 𝓠-homomorphism. Then S(Q1)/kerf Q S(Q1)f.

Proof

It is easy to see that the mapping ϕ: S(Q1)/kerfS(Q1)f is an isomorphism. Since f is a 𝓠-homomorphism, we have

(Q1kerf)ϕ=Q1f=Q2S(Q1)f=Q2(S(Q1)kerf)ϕ.

Thus ϕ is also a 𝓠-homomorphism and S(Q1)/kerf Q S(Q1)f. □

Theorem 4.7

If there exists a unitary surjective 𝓠-subhomomorphism φ of S(Q) onto G, then T(Q1) = π(S, G, φ) is an E-unitary 𝓠-regular cover of S(Q) through G.

Proof

By Theorem 4.4, T(Q1) = π(S, G, φ) is an E-unitary 𝓠-regular cover of S(Q). We prove that T/σ Q G as follow.

It follows from Theorem 3.7 that T(Q1) is a 𝓠-subdirect product of S(Q) and G. Hence G is the 𝓠-homomorphic image of T(Q1) under the projection f2 : T(Q1) → G, (a, g) ↦ g. Since G is a group, ker f2 is a group congruence on T(Q1), and so σkerf2. Conversely, if (a, g) ∈ (b, g)kerf2, then (a, g)f2 = (b, h)f2, so that g = h. Since T(Q1) is 𝓠-regular, for any (b, g) = (b, h) ∈ T(Q1) there exists (x, y) ∈ VQ1(b, g) such that

(b,g)(x,y)=(bx,gy)ET{(e,1)T:eES}.

Hence bxES and gy = 1, that is g−1 = y. So (x, g−1) ∈ T(Q1) and (bx, 1) ∈ Q1. Now (xa, 1) = (x, y)(a, g) ∈ T. Hence, by the definition of T, 1 ∈ (xa)φ. Notice that φ is unitary, we have xaES. Thus (xa, 1) ∈ ET, and so

(bx,1)(a,g)=(bxa,g)=(b,h)(xa,1).

Since (bx, 1), (xa, 1) ∈ ETU, by Lemma 4.1, (a, g)σ (b, h). Thus ker f2σ. Hence σ = kerf2. Therefore, it follows from lemma 4.6 that T/σ = T/kerf2 Q G. □

Remark 4.8

As we know, the class of 𝓠-regular semigroups properly contains the class of V-regular semigroups. In fact, if we restrict the C-set Q of a 𝓠-regular semigroup S to the set of idempotents, then the subdirect products of V-regular semigroups can be constructed in a similar way and we can also use this contruction to study the E-unitary cover for V-regular semigroups.

Acknowledgement

Research Supported by Fundamental Research Funds for the Central Universities.

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About the article

Received: 2016-11-01

Accepted: 2018-03-28

Published Online: 2018-05-30


Citation Information: Open Mathematics, Volume 16, Issue 1, Pages 522–530, ISSN (Online) 2391-5455, DOI: https://doi.org/10.1515/math-2018-0048.

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© 2018 Feng, published by De Gruyter. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License. BY-NC-ND 4.0

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