A construction of regular semigroups with quasiideal regular *-transversals
Let $S$ be a semigroup and let “$\ast$ ” be a unary operation on S satisfying the following identities: $$xx^{\ast} x = x, x^{\ast} xx^{\ast} = x^{\ast},\; x^{\ast \ast \ast} = x^{\ast},\; (xy^{\ast} )^{\ast} = y^{\ast \ast} x^{\ast},\; (x^{\ast} y)^{\ast} = y^{\ast} x^{\ast \ast}.$$ Then S\ast...
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Institute of Mathematics, NAS of Ukraine
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| author | Wang, Shou-Feng Ван, Шо-Фен |
| author_facet | Wang, Shou-Feng Ван, Шо-Фен |
| author_sort | Wang, Shou-Feng |
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| datestamp_date | 2019-12-05T09:32:19Z |
| description | Let $S$ be a semigroup and let “$\ast$ ” be a unary operation on S satisfying the following identities:
$$xx^{\ast} x = x, x^{\ast} xx^{\ast} = x^{\ast},\; x^{\ast \ast \ast} = x^{\ast},\; (xy^{\ast} )^{\ast} = y^{\ast \ast} x^{\ast},\; (x^{\ast} y)^{\ast} = y^{\ast} x^{\ast \ast}.$$
Then S\ast = \{ x\ast | x \in S\} is called a regular \ast -transversal of $S$ in the literatures.
We propose a method for the construction
of regular semigroups with quasiideal regular $\ast$ -transversals based on the use of fundamental regular semigroups and regular
$\ast$ -semigroups. |
| first_indexed | 2026-03-24T02:15:39Z |
| format | Article |
| fulltext |
UDC 512.5
Shou-Feng Wang (Yunnan Normal Univ., China)
A CONSTRUCTION OF REGULAR SEMIGROUPS
WITH QUASIIDEAL REGULAR \ast -TRANSVERSALS*
ПОБУДОВА РЕГУЛЯРНИХ НАПIВГРУП
IЗ КВАЗIIДЕАЛЬНИМИ РЕГУЛЯРНИМИ \ast -ТРАНВЕРСАЛЯМИ
Let S be a semigroup and let “\ast ” be a unary operation on S satisfying the following identities:
xx\ast x = x, x\ast xx\ast = x\ast , x\ast \ast \ast = x\ast , (xy\ast )\ast = y\ast \ast x\ast , (x\ast y)\ast = y\ast x\ast \ast .
Then S\ast = \{ x\ast | x \in S\} is called a regular \ast -transversal of S in the literatures. We propose a method for the construction
of regular semigroups with quasiideal regular \ast -transversals based on the use of fundamental regular semigroups and regular
\ast -semigroups.
Нехай S — напiвгрупа, а “\ast ” — унарна операцiя на S, що задовольняє такi тотожностi:
xx\ast x = x, x\ast xx\ast = x\ast , x\ast \ast \ast = x\ast , (xy\ast )\ast = y\ast \ast x\ast , (x\ast y)\ast = y\ast x\ast \ast .
Тодi S\ast = \{ x\ast | x \in S\} має в лiтературi назву регулярної \ast -трансверсалi S . Запропоновано новий метод побу-
дови регулярних напiвгруп з квазiiдеальними регулярними \ast -трансверсалями з використанням фундаментальних
регулярних напiвгруп та регулярних \ast -напiвгруп.
1. Introduction. Let S be a semigroup. We denote the set of all idempotents of S by E(S) and the
set of all inverses of x \in S by V (x). Recall that
V (x) = \{ a \in S | xax = x, axa = a\}
for any x \in S. A semigroup S is called regular if V (x) \not = \varnothing for any x \in S, and a regular semigroup
S is called inverse if E(S) is a commutative subsemigroup of S, or equivalently, the cardinality of
V (x) is equal to 1 for any x in S.
Recall from Petrich and Reilly [11] that a unary semigroup is a (2,1)-algebra (S, \cdot ,\ast ) where (S, \cdot ),
is a semigroup and the mapping a \mapsto \rightarrow a\ast is a unary operation on S. For brevity, we denote (S, \cdot ,\ast )
by (S, \ast ). It is well known that a regular semigroup S is inverse if and only if there exists a unary
operation “\ast ” on S satisfying the following identities:
xx\ast x = x, (x\ast )\ast = x, (xy)\ast = y\ast x\ast , xx\ast yy\ast = yy\ast xx\ast . (1.1)
Thus, inverse semigroups can be regarded as a class of unary semigroups.
Inspired by the above identity (1.1), regular \ast -semigroups were introduced in [10]. Recall that a
unary semigroup (S, \ast ) is called a regular \ast -semigroup if the following identities are satisfied:
xx\ast x = x, (x\ast )\ast = x, (xy)\ast = y\ast x\ast . (1.2)
Obviously, the class of regular \ast -semigroups forms a class of unary semigroups and contains the class
of inverse semigroups as a subclass. Regular \ast -semigroups are investigated in many papers (see, for
example, [5, 6, 10, 18, 19]).
* This paper is supported jointly by the Nature Science Foundations of China (№ 11661082, 11301470) and the Nature
Science Foundation of Yunnan Province (№ 2012FB139).
c\bigcirc SHOU-FENG WANG, 2016
1552 ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 11
A CONSTRUCTION OF REGULAR SEMIGROUPS WITH QUASIIDEAL REGULAR \ast -TRANSVERSALS 1553
On the other hand, Blyth and McFadden [1] introduced the concept of inverse transversals for
regular semigroups. A subsemigroup S\circ of a semigroup S is called an inverse transversal of S
if V (x) \cap S\circ contains one element exactly for all x \in S. Clearly, in this case, S\circ is an inverse
subsemigroup of S. From the remarks following Theorem 2 in Tang [12] and Theorem 4.8 in Tang
[13], we can deduce easily that a regular semigroup S contains an inverse transversal if and only if
there exists a unary operation “\ast ” on S satisfying the following identities:
xx\ast x = x, x\ast xx\ast = x\ast , x\ast \ast \ast = x\ast , (x\ast y)\ast = y\ast x\ast \ast ,
(xy\ast )\ast = y\ast \ast x\ast , x\ast x\ast \ast y\ast y\ast \ast = y\ast y\ast \ast x\ast x\ast \ast .
(1.3)
In this case, S\circ = \{ x\ast | x \in S\} is an inverse transversal of S. Therefore, the class of regular
semigroups with inverse transversals is a class of unary semigroups which also contains the class of
inverse semigroups as a subclass. Inverse transversals of regular semigroups are studied extensively
(see, for example, [1 – 3, 12, 13]).
Now, let (S, \ast ) be a unary semigroup and the unary operation “\ast ” satisfy the following identities:
xx\ast x = x, x\ast xx\ast = x\ast , x\ast \ast \ast = x\ast , (xy\ast )\ast = y\ast \ast x\ast , (x\ast y)\ast = y\ast x\ast \ast . (1.4)
Then S\ast = \{ x\ast | x \in S\} is called a regular \ast -transversal of S from Li [8]. Clearly, (S\ast , \ast ) is
a regular \ast -semigroup in this case. Moreover, combining the facts (1.2) and (1.3), we can see that
regular semigroups having regular \ast -transversals are generalizations of regular \ast -semigroups and
regular semigroups with inverse transversals.
Regular \ast -transversals have received serious attention in the literatures (see, e.g., [7 – 9, 14 – 16]).
Recently, the author initiated the investigations of regular semigroups with regular \ast -transversals by
fundamental approaches in Wang [17] in which fundamental regular semigroups with a quasiideal
regular \ast -transversal are constructed and a fundamental representation of any regular semigroup with
quasiideal regular \ast -transversals is obtained. Recall from Howie [4] that a semigroup is fundamental
if its maximum idempotent-separating congruence is the identity congruence.
In this paper, we shall continue to study regular semigroups with regular \ast -transversals by
fundamental approaches. After giving some necessary preliminaries, we give a construction method
of regular semigroups with quasiideal regular \ast -transversals by using regular \ast -semigroups and
fundamental regular semigroups constructed in Wang [17].
2. Preliminaries. Let (S, \ast ) be a regular \ast -semigroup. Then we write (S, \ast ) \in \bfr and
FS = \{ e \in E(S) | e\ast = e\} , and call FS the set of projections of (S, \ast ). It is easy to see that
FS = \{ xx\ast | x \in S\} = \{ x\ast x | x \in S\} . On regular \ast -semigroups, we have the following basic
results.
Lemma 2.1 [10, 18]. Let (S, \ast ) \in \bfr . Then
(1) (\forall e, f \in FS) ef \in FS =\Rightarrow ef = fe \in FS ;
(2) (\forall x \in S) x \in E(S) \Leftarrow \Rightarrow x\ast \in E(S);
(3) (FS)
2 \subseteq E(S) and xFSx
\ast , x\ast FSx \subseteq FS for all x \in S.
Now, let (S, \ast ) be a unary semigroup and S\ast be a regular \ast -transversal of S. Then we write
(S, \ast ) \in rt. Thus, (S\ast , \ast ) \in r if (S, \ast ) \in rt. A quasiideal of a semigroup S is a subsemigroup
T of S which satisfies that TST \subseteq T. If (S, \ast ) \in rt and S\ast is a quasiideal of S, then we write
(S, \ast ) \in qit. In this case, we denote IS = \{ aa\ast | a \in S\} and \Lambda S = \{ a\ast a | a \in S\} .
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 11
1554 SHOU-FENG WANG
Lemma 2.2 (Lemmas 4.1 and 4.2 in [8], Corollary 2.5 in [17]). Let (S, \ast ) \in \bfq \bfi \bft . Then
(1) IS = \{ e \in E(S) | e\scrL e\ast \} , \Lambda S = \{ f \in E(S) | f\scrR f\ast \} and FS\ast = IS \cap \Lambda S ;
(2) g\ast \ast = g\ast \in FS\ast for all g \in IS \cup \Lambda S ;
(3) fg \in S\ast and so fg = (fg)\ast \ast for all f \in \Lambda S and g \in IS ;
(4) (xy)\ast \ast = x\ast \ast x\ast xyy\ast y\ast \ast for all x, y \in S.
Now, let (S, \ast ) \in \bfq \bfi \bft and e \in IS , f \in \Lambda S . Denote
\langle e\rangle = eISe = \{ eie | i \in IS\} , \langle f\rangle = f\Lambda Sf = \{ f\lambda f | \lambda \in \Lambda S\} .
Lemma 2.3 (Lemma 2.10 in [17]). Let (S, \ast ) \in \bfq \bfi \bft , a \in S, e \in IS , f \in \Lambda S and p \in FS\ast .
Then \langle p\rangle \subseteq FS\ast and
(1) \langle e\rangle = eFS\ast e\ast = \{ x \in IS | exe = x\} and \langle f\rangle = f\ast FS\ast f = \{ x \in \Lambda S | fxf = x\} ;
(2) xyx \in \langle e\rangle for all x, y \in \langle e\rangle and xyx \in \langle f\rangle for all x, y \in \langle f\rangle ;
(3) a\ast xa \in \langle a\ast a\rangle for all x \in \langle aa\ast \rangle and aya\ast \in \langle aa\ast \rangle for all y \in \langle a\ast a\rangle .
Recall that a semiband is a semigroup which is generated by its idempotents. Let C be a regular
semiband, (C, \ast ) \in \bfq \bfi \bft and use I and \Lambda to denote IC and \Lambda C , respectively. In view of Lemma 2.3,
we have xyx \in \langle e\rangle for all x, y \in \langle e\rangle and e \in I \cup \Lambda .
Now, let e, f \in I \cup \Lambda . A bijection \alpha from \langle e\rangle onto \langle f\rangle is called a pre-isomorphism if\bigl(
\forall x, y \in \langle e\rangle
\bigr)
(xyx)\alpha = (x\alpha )(y\alpha )(x\alpha ). (2.1)
Clearly, e\alpha = f in the case. Moreover, we say that \langle e\rangle and \langle f\rangle are pre-isomorphic if there exists
a pre-isomorphism from \langle e\rangle onto \langle f\rangle . In this case, we write \langle e\rangle \simeq \langle f\rangle and denote the set of all
pre-isomorphisms from \langle e\rangle onto \langle f\rangle by Te,f . The following result shows that pre-isomorphisms exist
indeed. As usual, we use \iota M to denote the identity transformation on the nonempty set M.
Lemma 2.4 (Proposition 3.1 in [17]). Let C be a regular semiband and (C, \ast ) \in \bfq \bfi \bft . Define
\pi a : \langle aa\ast \rangle \rightarrow \langle a\ast a\rangle , x \mapsto \rightarrow a\ast xa.
Then \pi a \in Taa\ast ,a\ast a. Moreover, the inverse mapping of \pi a is
\pi - 1
a : \langle a\ast a\rangle \rightarrow \langle aa\ast \rangle , y \mapsto \rightarrow aya\ast
and \pi - 1
a \in Ta\ast a,aa\ast . In particular, we have \pi p = \iota \langle p\rangle = \pi - 1
p for any p \in FC\ast .
On pre-isomorphisms in general, we have the following results.
Lemma 2.5 (Lemma 3.2 in [17]). Let e \in I, f \in \Lambda , x \in \langle e\rangle , y \in \langle f\rangle and \alpha \in Te,f . Then
(1) \alpha - 1 \in Tf,e;
(2) \langle x\rangle \alpha = \langle x\alpha \rangle , \langle y\rangle \alpha - 1 = \langle y\alpha - 1\rangle ;
(3) (x\alpha )\ast = (x\alpha )f\ast , x\alpha = (x\alpha )\ast f ;
(4) (y\alpha - 1)\ast = e\ast (y\alpha - 1), y\alpha - 1 = e(y\alpha - 1)\ast .
Denote \scrU = \{ (e, f) \in I \times \Lambda | \langle e\rangle \simeq \langle f\rangle \} and define a multiplication “\circ ” on the set
TC =
\bigcup
(e,f)\in \scrU
Te,f
as follows: for \alpha \in Te,f and \beta \in Tg,h,
\alpha \circ \beta = \alpha \pi - 1
g(fg)\ast f\beta , \alpha \ast = \pi f\alpha
- 1\pi e,
where the composition is the that in the symmetric inverse semigroup on the set I \cup \Lambda .
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 11
A CONSTRUCTION OF REGULAR SEMIGROUPS WITH QUASIIDEAL REGULAR \ast -TRANSVERSALS 1555
Lemma 2.6 (Lemma 3.3, Theorem 3.5, Corollary 3.9 and Theorem 3.10 in [17]). With the above
notations, we have the following results:
(1) (TC ,
\ast ) \in \bfq \bfi \bft , TC is fundamental and T \ast
C = \{ \alpha \in TC | \alpha \in Tp,q, p, q \in FC\ast \} ;
(2) if \alpha , \beta \in TC and \alpha \in Te,f , \beta \in Tg,h, then \alpha \circ \beta \in Tj,k, where j = (fg(fg)\ast f)\alpha - 1,
k = (g(fg)\ast fg)\beta ;
(3) \alpha \ast \in Tf\ast ,e\ast , \alpha
\ast \ast \in Te\ast ,f\ast , \alpha \circ \alpha \ast = \pi e, \alpha
\ast \circ \alpha = \pi f .
In the rest of this section, we let (S, \ast ) \in \bfq \bfi \bft and C be the semiband generated by E(S). Then
we have the following lemma.
Lemma 2.7 (Lemma 4.1 in [17]). Let (S, \ast ) \in \bfq \bfi \bft . Then (C, \ast ) \in \bfq \bfi \bft . In this case, C\ast =
= C \cap S\ast , IS = IC , \Lambda S = \Lambda C and FS\ast = FC\ast .
By Lemmas 2.6 and 2.7, we can construct a fundamental regular semigroup TC with a quasiideal
regular \ast -transversal T \ast
C . For a \in S, by Lemma 2.3 (3), we can define
\rho a : \langle aa\ast \rangle \rightarrow \langle a\ast a\rangle , x \mapsto \rightarrow a\ast xa.
Then the inverse mapping \rho - 1
a of \rho a is
\rho - 1
a : \langle a\ast a\rangle \rightarrow \langle aa\ast \rangle , y \mapsto \rightarrow aya\ast .
Observe that \rho a = \pi a for every a \in C where \pi a is defined as in Lemma 2.4. Moreover, we also
need the following result.
Lemma 2.8 (Lemma 4.2 and Theorem 4.3 in [17]). Let a, b \in S. Then
(1) \rho a \in Taa\ast ,a\ast a and \rho - 1
a \in Ta\ast a,aa\ast ;
(2) \rho a \circ \rho b = \rho ab and (\rho a)
\ast = \rho a\ast in TC .
3. Main result. In this section, a structure theorem of regular semigroups with a quasiideal
regular \ast -transversal is obtained by using a fundamental regular semigroup and a regular \ast -semigroup.
Let C be a semiband, (C, \ast ) \in \bfq \bfi \bft and (R, \ast ) \in r. Assume that (C\ast , \ast ) is a common
(2,1)-subalgebra of (R, \ast ) and (C, \ast ) such that R \cap C = C\ast and FR = FC\ast . By Lemma 2.6 (1),
we have (TC ,
\ast ) \in qit and TC is fundamental.
Now, let a \in R. Then a\ast xa \in a\ast aFRa
\ast a for every x in aa\ast FRaa
\ast by Lemma 2.1 (3). Therefore,
we can define a mapping as follows:
\lambda a : aa\ast FRaa
\ast \rightarrow a\ast aFRa
\ast a, x \mapsto \rightarrow a\ast xa.
It can be proved easily that xyx \in aa\ast FRaa
\ast and
(xyx)\lambda a = (x\lambda a)(y\lambda a)(x\lambda a) (3.1)
for all x, y \in aa\ast FRaa
\ast .
Lemma 3.1. With the above notations, the following statements hold for all a, b \in R:
(1) \lambda a \in Taa\ast ,a\ast a \subseteq T \ast
C . In particular, if a \in C\ast , then \lambda a = \pi a where \pi a is defined as in
Lemma 2.4;
(2) (\lambda a)
\ast = \lambda a\ast and \lambda a \circ \lambda b = \lambda ab in T \ast
C where ab is taken in R.
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 11
1556 SHOU-FENG WANG
Proof. (1) Since FR = FC\ast and aa\ast , a\ast a \in FR, we have aa\ast , a\ast a \in FC\ast = I \cap \Lambda and
(aa\ast )\ast = aa\ast , (a\ast a)\ast = a\ast a in C. By Lemma 2.3 (1), we obtain
\langle aa\ast \rangle = aa\ast FC\ast aa\ast = aa\ast FRaa
\ast , \langle a\ast a\rangle = a\ast aFC\ast a\ast a = a\ast aFRa
\ast a
in C. This implies that \mathrm{d}\mathrm{o}\mathrm{m}\lambda a = \langle aa\ast \rangle and \mathrm{r}\mathrm{a}\mathrm{n}\lambda a = \langle a\ast a\rangle . Moreover, it is easy to see that \lambda a is
bijective. In fact, the inverse mapping \lambda - 1
a of \lambda a is
\lambda a\ast : \langle a\ast a\rangle \rightarrow \langle aa\ast \rangle , y \mapsto \rightarrow aya\ast .
Combining identities (2.1) and (3.1), we can obtain that \lambda a \in Taa\ast ,a\ast a. If a \in C\ast , then the product
a\ast xa can be taken both in C and in R for all x \in \langle aa\ast \rangle and thus we have \lambda a = \pi a where \pi a is
defined as in Lemma 2.4.
(2) Since aa\ast , a\ast a \in FR = FC\ast , we have \pi a\ast a = \iota \langle a\ast a\rangle and \pi aa\ast = \iota \langle aa\ast \rangle by Lemma 2.4. This
implies that
(\lambda a)
\ast = \pi a\ast a\lambda
- 1
a \pi aa\ast = \iota \langle a\ast a\rangle \lambda
- 1
a \iota \langle aa\ast \rangle = \lambda - 1
a = \lambda a\ast
in TC . On the other hand, since a\ast a, bb\ast \in FR = FC\ast , we have
\lambda a \circ \lambda b = \lambda a\pi
- 1
bb\ast (a\ast abb\ast )\ast a\ast a\lambda b = \lambda a\pi
- 1
bb\ast bb\ast a\ast aa\ast a\lambda b = \lambda a\pi
- 1
bb\ast a\ast a\lambda b = \lambda a\lambda
- 1
bb\ast a\ast a\lambda b
by item (1) and the fact that (bb\ast )(a\ast a) \in C\ast , and
(a\ast abb\ast a\ast a)\lambda - 1
a = a(a\ast abb\ast a\ast a)a\ast = ab(ab)\ast , (bb\ast a\ast abb\ast )\lambda b = b\ast (bb\ast a\ast abb\ast )b = (ab)\ast ab
whence \lambda a \circ \lambda b \in Tab(ab)\ast ,(ab)\ast ab by Lemma 2.6 (2). Moreover, it follows that
x(\lambda a \circ \lambda b) = x(\lambda a\lambda
- 1
bb\ast a\ast a\lambda b) = b\ast (bb\ast a\ast a(a\ast xa)(bb\ast a\ast a)\ast )b = b\ast a\ast xab = (ab)\ast xab = x\lambda ab
for all x \in \mathrm{d}\mathrm{o}\mathrm{m}(\lambda a \circ \lambda b) = \mathrm{d}\mathrm{o}\mathrm{m}\lambda ab. This implies that \lambda a \circ \lambda b = \lambda ab.
Lemma 3.1 is proved.
Now, let
W = \{ (\alpha , a) \in TC \times R | \alpha \ast \ast = \lambda a\}
and define a binary operation and a unary operation “\ast ” as follows: for \alpha \in Te,f , \beta \in Tg,h and
(\alpha , a), (\beta , b) \in W,
(\alpha , a)(\beta , b) =
\bigl(
\alpha \circ \beta , a(fg)b
\bigr)
, (\alpha , a)\ast = (\alpha \ast , a\ast ),
where fg \in C\ast \subseteq R by Lemma 2.2 (3) and the product a(fg)b is taken in R.
Theorem 3.1. With the above notations, (W, \ast ) \in \bfq \bfi \bft . Conversely, any (S, \ast ) \in \bfq \bfi \bft can be
constructed in this way.
Proof. The binary operation and the unary operation are well-defined. In fact, let \alpha \in Te,f ,
\beta \in Tg,h and (\alpha , a), (\beta , b) \in W. Then \alpha \ast \ast = \lambda a and \beta \ast \ast = \lambda b. In view of Lemma 2.2 (4), we have
(\alpha \circ \beta )\ast \ast = \alpha \ast \ast \circ (\alpha \ast \circ \alpha \circ \beta \circ \beta \ast ) \circ \beta \ast \ast = \lambda a \circ (\pi f \circ \pi g) \circ \lambda b = \lambda a \circ \pi fg \circ \lambda b
by Lemmas 2.6 (3) and 2.8. Since fg \in C \cap R = C\ast by Lemma 2.2 (3), we have \pi fg = \lambda fg by
Lemma 3.1 (1), this shows that
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 11
A CONSTRUCTION OF REGULAR SEMIGROUPS WITH QUASIIDEAL REGULAR \ast -TRANSVERSALS 1557
(\alpha \circ \beta )\ast \ast = \lambda a \circ \lambda fg \circ \lambda b = \lambda a(fg)b
by Lemma 3.1 (2) and so
(\alpha , a)(\beta , b) =
\bigl(
\alpha \circ \beta , a(fg)b
\bigr)
\in W.
On the other hand, observe that (\alpha \ast )\ast \ast = (\alpha \ast \ast )\ast = (\lambda a)
\ast = \lambda a\ast by Lemma 3.1 (2) again, it follows
that (\alpha , a)\ast = (\alpha \ast , a\ast ) \in W.
The above binary operation is associative. In fact, let
\alpha \in Te,f , \beta \in Tg,h, \gamma \in Ts,t, \alpha \circ \beta \in Tj,k, \beta \circ \gamma \in Tp,q, (\alpha , a), (\beta , b), (\gamma , c) \in W,
where k = (g(fg)\ast fg)\beta by Lemma 2.6 (2). By Lemma 2.5 (3), we have k = k\ast h and so
ks = k\ast (hs). Since (\alpha \circ \beta , a(fg)b) \in W and \alpha \circ \beta \in Tj,k, it follows that
Tj\ast ,k\ast \ni (\alpha \circ \beta )\ast \ast = \lambda a(fg)b \in Ta(fg)b(a(fg)b)\ast ,(a(fg)b)\ast a(fg)b
by Lemma 2.6 (3), whence k\ast = (a(fg)b)\ast a(fg)b. Thus, we get
(a(fg)b)(ks)c = (a(fg)b)k\ast (hs)c =
\bigl(
a(fg)b
\bigr)
\cdot
\bigl(
a(fg)b
\bigr) \ast
a(fg)b) \cdot (hs)c = a(fg)b(hs)c.
Dually, we can prove that a(fp)(b(hs)c) = a(fg)b(hs)c. Thus,\bigl[
(\alpha , a)(\beta , b)
\bigr]
(\gamma , c) =
\bigl(
(\alpha \circ \beta ) \circ \gamma , (a(fg)b)(ks)c
\bigr)
=
=
\bigl(
\alpha \circ (\beta \circ \gamma ), a(fp)(b(hs)c)
\bigr)
= (\alpha , a)
\bigl[
(\beta , b)(\gamma , c)
\bigr]
.
Let \alpha \in Te,f , \beta \in Tg,h and (\alpha , a), (\beta , b) \in W. Then
\alpha \ast \in Tf\ast ,e\ast , \beta \ast \in Th\ast ,g\ast Te\ast ,f\ast \ni \alpha \ast \ast = \lambda a \in Taa\ast ,a\ast a, Tg\ast ,h\ast \ni \beta \ast \ast = \lambda b \in Tbb\ast ,b\ast b (3.2)
by Lemmas 2.6 (3) and 3.1 (1). This implies that e\ast = aa\ast and f\ast = a\ast a. Thus, we have
(\alpha , a)(\alpha , a)\ast (\alpha , a) = (\alpha , a)(\alpha \ast , a\ast )(\alpha , a) = (\alpha \circ \alpha \ast \circ \alpha , a(ff\ast )a\ast (e\ast e)a) = (\alpha , a).
Similarly, we can see that (\alpha , a)\ast (\alpha , a)(\alpha , a)\ast = (\alpha , a)\ast . On the other hand, observe that (fh\ast )\ast =
= h\ast \ast f\ast = h\ast f\ast by (1.4), it follows that\bigl[
(\alpha , a)(\beta , b)\ast
\bigr] \ast
=
\bigl[
(\alpha , a)(\beta \ast , b\ast )
\bigr] \ast
=
\bigl(
(\alpha \circ \beta \ast )\ast , b\ast \ast (fh\ast )\ast a\ast
\bigr)
=
=
\bigl(
\beta \ast \ast \circ \alpha \ast , b\ast \ast (h\ast f\ast )a\ast
\bigr)
= (\beta \ast \ast , b\ast \ast )(\alpha \ast , a\ast ) = (\beta , b)\ast \ast (\alpha , a)\ast .
Similarly, we can see that
\bigl[
(\alpha , a)\ast (\beta , b)
\bigr] \ast
= (\beta , b)\ast (\alpha , a)\ast \ast .
Recall that T \ast
C = \{ \alpha \in TC | \alpha \in Tp,q, p, q \in FC\ast \} by Lemma 2.6 (1). We assert that W \ast =
= \{ (\alpha , a) \in W | \alpha \in T \ast
C\} . Obviously, W \ast \subseteq \{ (\alpha , a) \in W | \alpha \in T \ast
C\} . On the other hand, if
(\alpha , a) \in W and \alpha \in T \ast
C , then \alpha = \alpha \ast \ast , a\ast \ast = a and (\alpha \ast , a\ast ) \in W. This implies that (\alpha , a) =
= (\alpha \ast , a\ast )\ast \in W \ast . Thus, \{ (\alpha , a) \in W | \alpha \in T \ast
C\} \subseteq W \ast . Now, let (\alpha , a), (\gamma , c) \in W \ast and
(\beta , b) \in W. Since (TC ,
\ast ) \in \bfq \bfi \bft , \alpha \circ \beta \circ \gamma \in T \ast
C . This implies that (\alpha , a)(\beta , b)(\gamma , c) \in W \ast and so
(W, \ast ) \in \bfq \bfi \bft .
Conversely, let (S, \ast ) \in \bfq \bfi \bft and C be the semiband generated by E(S). Then (C, \ast ) \in \bfq \bfi \bft ,
(S\ast , \ast ) is a regular \ast -semigroup, and (C\ast , \ast ) is a (2,1)-subalgebra of (S\ast , \ast ) and (C, \ast ) such that
S\ast \cap C = C\ast and FS\ast = FC\ast by Lemma 2.7. By the direct part, we have a semigroup
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 11
1558 SHOU-FENG WANG
W = \{ (\alpha , a) \in TC \times S\ast | \alpha \ast \ast = \lambda a\}
and (W, \ast ) \in \bfq \bfi \bft with respect to
(\alpha , a)(\beta , b) =
\bigl(
\alpha \circ \beta , a(fg)b
\bigr)
, (\alpha , a)\ast = (\alpha \ast , a\ast ),
where (\alpha , a), (\beta , b) \in W, \alpha \in Te,f , \beta \in Tg,h and fg \in C\ast by Lemma 2.2 (3). Observe that \lambda a = \rho a
where \rho a is defined as in the statements before Lemma 2.8 for all a \in S\ast .
In what follows, we prove that
\psi : S \rightarrow W, x \mapsto \rightarrow (\rho x, x
\ast \ast )
is a unary isomorphism through the following steps where \rho x is defined as in the statements before
Lemma 2.8.
(1) By Lemma 2.8 (2), we have (\rho x)
\ast = \rho x\ast and so (\rho x)
\ast \ast = \rho x\ast \ast = \lambda x\ast \ast and
(x\psi )\ast = (\rho x, x
\ast \ast )\ast = ((\rho x)
\ast , x\ast \ast \ast ) = (\rho x\ast , x\ast \ast \ast ) = x\ast \psi .
This implies that \psi is well-defined and preserves the unary operation “\ast .”
(2) Since \rho x \in Txx\ast ,x\ast x, \rho y \in Tyy\ast ,y\ast y by Lemma 2.8 (1), we have
(xy)\psi = (\rho xy, (xy)
\ast \ast ) = (\rho x \circ \rho y, x\ast \ast (x\ast xyy\ast )y\ast \ast ) = (\rho x, x
\ast \ast )(\rho y, y
\ast \ast )
by Lemma 2.2 (4) and Lemma 2.8 (2).
(3) If x, y \in S and (\rho x, x
\ast \ast ) = (\rho y, y
\ast \ast ), then \rho x = \rho y, x
\ast \ast = y\ast \ast . This implies that
Txx\ast ,x\ast x \ni \rho x = \rho y \in Tyy\ast ,y\ast y
by Lemma 2.8 (1) and so
xx\ast = yy\ast , x\ast x = y\ast y, x\ast \ast = y\ast \ast .
Thus, x = xx\ast x\ast \ast x\ast x = yy\ast y\ast \ast y\ast y = y.
(4) If (\alpha , a) \in W and \alpha \in Te,f , then \alpha \ast \ast = \lambda a = \rho a since a \in S\ast . By Lemma 2.6 (3),
Lemma 2.8 and the fact that e, f \in C, we obtain
\alpha = \alpha \circ \alpha \ast \circ \alpha \ast \ast \circ \alpha \ast \circ \alpha = \pi e \circ \rho a \circ \pi f = \rho e \circ \rho a \circ \rho f = \rho eaf .
This shows that (\alpha , a) = (\rho eaf , a). Since (\alpha , a) \in W and \alpha \in Te,f , we have a \in S\ast , e\ast = aa\ast and
f\ast = a\ast a by (3.2). This implies that
(eaf)\ast \ast = (ea)\ast \ast ((ea)\ast eaff\ast )f\ast \ast = e\ast aa\ast e\ast eaff\ast = a
by Lemma 2.2 (4) and item (1.4). Therefore (eaf)\psi = (\rho eaf , (eaf)
\ast \ast ) = (\alpha , a).
Theorem 3.1 is proved.
We end our paper by giving the following example.
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 11
A CONSTRUCTION OF REGULAR SEMIGROUPS WITH QUASIIDEAL REGULAR \ast -TRANSVERSALS 1559
Example 3.1. Let S be a completely simple semigroup and H be an \scrH -class of S. Then H is
a group and contains exactly one inverse of a for any a \in S. Denote the identity of H by e\circ and
the unique inverse of a in H by a\circ for a \in S. Then H is an inverse transversal of S such that
HSH \subseteq H and so (S, \ast ) \in qit with the operation “\ast ” defined by a\ast = a\circ for any a \in S. Obviously,
H = S\ast and FS\ast = \{ e\circ \} in the case. Consider C = \langle E(S)\rangle . Then (C, \ast ) \in \bfq \bfi \bft , C\ast = H \cap C and
FC\ast = \{ e\circ \} . Moreover,
I = IC = IS = \{ e \in E(S) | e\scrL e\circ \} , \Lambda = \Lambda C = \Lambda S = \{ f \in E(S) | f\scrR e\circ \}
by Lemmas 2.7 and 2.2 (1). For any a \in S\ast = H, the mapping
\lambda a : \langle aa\ast \rangle = \{ e\circ \} \rightarrow \langle a\ast a\rangle = \{ e\circ \} , x \mapsto \rightarrow a\ast xa
is always \iota \{ e\circ \} . On the other hand, for any e \in I and f \in \Lambda , we have \langle e\rangle = \{ e\} and \langle f\rangle = \{ f\} .
Denote
\sigma e,f : \langle e\rangle \rightarrow \langle f\rangle , e \mapsto \rightarrow f, e \in I, f \in \Lambda .
Then Te,f = \{ \sigma e,f\} for all e \in I and f \in \Lambda and so TC = \{ \sigma e,f | e \in I, f \in \Lambda \} . By Lemma 2.6 (2),
\sigma e,f \circ \sigma g,h \in T(fg(fg)\ast f)\sigma - 1
e,f ,(g(fg)
\ast fg)\sigma g,h
= Tf\sigma - 1
e,f ,g\sigma g,h
= Te,h
for all \sigma e,f , \sigma g,h \in TC . This implies that
\sigma e,f \circ \sigma g,h = \sigma e,h, \sigma \ast e,f = \iota \{ e\circ \}
for all e, g \in I and f, h \in \Lambda . Thus, we can form the following semigroup:
W = \{ (\alpha , a) \in TC \times H | \alpha \ast \ast = \lambda a\}
with the operation
(\sigma e,f , a)(\sigma g,h, b) =
\bigl(
\sigma e,h, a(fg)b
\bigr)
,
where fg \in S\ast = H by Lemma 2.2 (3). Observe that \lambda a = \iota \{ e\circ \} = \sigma \ast \ast e,f for all a \in H and e \in I,
f \in \Lambda , it follows that W = TC \times H. It is routine to check that W is isomorphic to the semigroup
M = I \times H \times \Lambda with respect to the following binary operation:
(e, a, f)(g, b, h) = (e, a(fg)b, h).
By Theorem 3.1, S is isomorphic to M. However, M is just a Rees matrix semigroup over the group
H. Thus, we obtain the well-known Rees constructions of completely simple semigroups by applying
Theorem 3.1.
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Received 18.02.14
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 11
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| id | umjimathkievua-article-1941 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:15:39Z |
| publishDate | 2016 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/9e/ef6590dfcfe2b0504abee15fd609e39e.pdf |
| spelling | umjimathkievua-article-19412019-12-05T09:32:19Z A construction of regular semigroups with quasiideal regular *-transversals Ппобудова регулярних напiвгруп iз квазiiдеальними регулярними $\ast$-транверсалями Wang, Shou-Feng Ван, Шо-Фен Let $S$ be a semigroup and let “$\ast$ ” be a unary operation on S satisfying the following identities: $$xx^{\ast} x = x, x^{\ast} xx^{\ast} = x^{\ast},\; x^{\ast \ast \ast} = x^{\ast},\; (xy^{\ast} )^{\ast} = y^{\ast \ast} x^{\ast},\; (x^{\ast} y)^{\ast} = y^{\ast} x^{\ast \ast}.$$ Then S\ast = \{ x\ast | x \in S\} is called a regular \ast -transversal of $S$ in the literatures. We propose a method for the construction of regular semigroups with quasiideal regular $\ast$ -transversals based on the use of fundamental regular semigroups and regular $\ast$ -semigroups. Нехай $S$ — напiвгрупа, а “$\ast$ ” — унарна операцiя на $S$, що задовольняє такi тотожностi: $$xx^{\ast} x = x, x^{\ast} xx^{\ast} = x^{\ast},\; x^{\ast \ast \ast} = x^{\ast},\; (xy^{\ast} )^{\ast} = y^{\ast \ast} x^{\ast},\; (x^{\ast} y)^{\ast} = y^{\ast} x^{\ast \ast}.$$ Тодi $S^{\ast} = \{ x^{\ast} | x \in S\}$ має в лiтературi назву регулярної $\ast$ -трансверсалi $S$. Запропоновано новий метод побудови регулярних напiвгруп з квазiiдеальними регулярними $\ast$ -трансверсалями з використанням фундаментальних регулярних напiвгруп та регулярних $\ast$ -напiвгруп. Institute of Mathematics, NAS of Ukraine 2016-11-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/1941 Ukrains’kyi Matematychnyi Zhurnal; Vol. 68 No. 11 (2016); 1552-1560 Український математичний журнал; Том 68 № 11 (2016); 1552-1560 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/1941/923 Copyright (c) 2016 Wang Shou-Feng |
| spellingShingle | Wang, Shou-Feng Ван, Шо-Фен A construction of regular semigroups with quasiideal regular *-transversals |
| title | A construction of regular semigroups with quasiideal regular *-transversals |
| title_alt | Ппобудова регулярних напiвгруп iз квазiiдеальними регулярними $\ast$-транверсалями |
| title_full | A construction of regular semigroups with quasiideal regular *-transversals |
| title_fullStr | A construction of regular semigroups with quasiideal regular *-transversals |
| title_full_unstemmed | A construction of regular semigroups with quasiideal regular *-transversals |
| title_short | A construction of regular semigroups with quasiideal regular *-transversals |
| title_sort | construction of regular semigroups with quasiideal regular *-transversals |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/1941 |
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