Download THE JOIN OF THE VARIETIES OLBG AND ORBG 1. Introduction

SOOCHOW JOURNAL OF MATHEMATICS
Volume 22, No. 3, pp. 405-410, July 1996
THE JOIN OF THE VARIETIES OLBG AND ORBG
BY
HAIXUAN YANG
Abstract. In this paper the varieties OLBG and ORBG are investigated. And the join of the varieties of OLBG and ORBG is described.
1. Introduction
Completely regular semigroups are semigroups that are unions of their
subgroups. They may be regarded as universal algebras with an associative
binary operation (multiplication) and a unary operation (inversion). As universal algebras, completely regular semigroups form a variety determined by
the identities
x = xx;1x xx;1 = x;1x (x;1 );1 = x:
Let CR denote this variety and LCR denote the lattice of subvarieties of
CR.
In 1975, M. Petrich 1] showed that B _ G = OBG, where B is the variety
of bands, G is the variety of groups and OBG is the variety of orthodox bands
of groups. In 1980, T.E. Hall and P.R. Jones 2] showed that B _ CS = POBG,
where CS is the variety of completely simple semigroups and POBG is the
variety of pseudo-orthodox bands of groups. In 1985, N.R. Reilly 3] showed
that O _ CS = O+ \ PO, where O+ = fS 2 CR : S= 2 Og, is the maximum
Received August 28, 1995.
AMS Subject Classication. 20M07.
This work is supported by National Natural Science Foundation and Doctoral Research Program Foundation of Chinese Education Committee. The author wishes to thank
Professor Y. Q. Guo for his help.
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HAIXUAN YANG
idempotent-separating congruence, O is the variety of orthodox completely
regular semigroups. In this paper, we investigate the variety OLBG
ORBG]
of orthodox completely regular semigroups on which Green's relation H is
a left right] congruence, In Theorem 3.4, the join of OLBG and ORBG is
described.
We shall use the following notation.
L0: the largest congruence contained in L
R0: the largest congruence contained in R
: the maximum idempotent-separating congruence
: the maximum idempotent pure congruence
Is : the equality relation on a semigroup S u = v ] 2A : the variety of completely regular semigroups determined by the
identities u = v ( 2 A).
LBG
RBG]: the variety of completely regular semigroups on which Green's
relation H is a left right] congruence.
Let a0 = aa;1 = a;1 a. Then B = x2 = x], O = (x0 y0 )0 = x0 y0 ], and
SLG = x0y = yx0] is the variety of Cliord semigroups.
For undened notation or terminology see 4].
2. OLBG
Lemma 2.1. Let S 2 CR. Then S 2 O if and only if S= 2 SLG.
Proof. Let S 2 O v = f(a b) 2 S S : V (a) = V (b)g: Then S=v is an
inverse semigroup (see 4]), hence S=v is a Cliord semigroup, i.e. S=v 2 SLG.
Obviously v is idempotent pure and thus v . Therefore S= 2 SLG.
Conversely, Let S= 2 SLG e,f 2 E (S ). Then ef 2 E (S= ): By Lallement's Lemma (see 4]), there exists g 2 E (S ) such that (ef g) 2 . Since is idempotent pure, ef 2 E (S ): Therefore S is orthodox.
Obviously we have
Lemma 2.2. Let S be a semigroup. Then L0 .
THE JOIN OF THE VARIETIES OLBG AND ORBG
407
Now we can prove the following theorem:
Theorem 2.3. The following conditions on a completely regular semi-
group S are equivalent.
(I) S 2 OLBG.
(II) S=L0 2 B S= 2 SLG.
(III) S=L0 2 LBG S= 2 SLG.
(IV) S= 2 LBG S= 2 SLG.
(V) S satises the identities (xy)0 = (xy0 )0 (x0 y0 )0 = x0 y0 .
(VI) S satises the identity (x0 y)0 = x0 y0 .
Proof. (I) implies (IV). It follows directly from Lemma 2.1.
(IV) implies (III). It follows directly from Lemma 2.2.
(III) implies (II). Let S=L0 2 LBG. Let L01 be the largest congruence
contained in the Green's L-relation on S=L0 , H be the Green's H -relation
on S=L0 . Then H is a left congruence and thus H L01 . We may dene a
function from S to S=L0 =L01 by
: a ! aL01 (a 2 S )
where a = aL0 . Then is a surjective homomorphism. Since H L01 S=L0 =L01
2 B . Let be the kernel of . Let (a b) 2 , i.e. (a b) 2 L01 . Then there
exists x 2 S=L0 such that a = xb, i.e. (a xb) 2 L0 , and thus there exists y 2 S
such that a = yxb 2 Sb. Similarly b 2 Sa, hence aLb. So L and thus
L0. Therefore S=L0 2 B since S= 2 B .
(II) implies (V). The hypothesis implies that yL0 y0 , hence xyL0 xy0 . Since
xyRxy0, xyHxy0, i.e. (xy)0 = (xy0)0 .
Since S= 2 SLG, S 2 O by Lemma 2.1, and thus (x0 y0 )0 = x0 y0 .
(V)implies (VI). This is obvious.
(VI)implies (I). Let e, f 2 E (S ), then ef = (ef )0 2 E (S ), hence S 2 O.
Let x y z 2 S xHy. Then (z 0 x)0 = z 0 x0 = z 0 y0 = (z 0 y)0 . Hence zx =
zz0 x = zz0 x(z0 x)0 = zz0 x(z0 y)0 = zz0 x(z0y);1 z;1zy 2 Szy. Similarly zy 2
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HAIXUAN YANG
Szx and thus zxLzy. Obviously zxRzy and thus zxHzy. Therefore H is a
left congruence, as required.
In view of Theorem 2.3, OLBG is a variety and OLBG= (x0 y)0 = x0 y0 ].
Dually ORBG = (xy0 )0 = x0 y0 ].
3. OLBG _ ORBG
Notation 3.1. (
6]) Let V 2 LCR, dene
V T 1 = fS 2 CR : S=L0 2 V g
V Tr = fS 2 CR : S=R0 2 V g:
Lemma 3.2. (
5, lemma 2.4] 6]) Let V 2 LCR, and V = u = v ]
Then
V Tr = (u x)0 = (v xu x)0 (v x)0 = (u xv x)0 ]
V T 1 = (xu )0 = (xu xv )0 (xv )0 = (xv xu )0 ]
2A
.
2A
2A
:
Lemma 3.3. Let S be a completely regular semigroup. Then \ H = Is.
Proof. Let a b 2 He, e 2 E (S ), and (a b) 2 . Then (aa;1 ba;1 ) 2 .
Since is idempotent pure, ba;1 2 E (S ). Since a b 2 He, ba;1 2 He, and
thus ba;1 = e. Therefore a = b, as required.
Theorem 3.4. The following conditions on a completely regular semi-
group S are equivalent.
(I) S 2 OLBG _ ORBG.
(II) S is a subdirect product of a semigroup in OLBG and a semigroup in
ORBG.
(III) S 2 O \ LBGTr \ RBGT 1 .
(IV) S= 2 SLG, S=R0 2 LBG, S=L0 2 RBG.
Proof. (I) implies (III). By implication (I))(II) of Theorem 2.3 and
its dual, it is easily checked that OLBG O \ LBGTr \ RBGT 1 , ORBG THE JOIN OF THE VARIETIES OLBG AND ORBG
409
O \ LBGTr \ RBGT 1. Therefore OLBG _ ORBG O \ LBGTr \ RBGT 1 , as
required.
(III) implies (IV). This follows directly from Notation 3.1 and Lemma 2.1.
(IV) implies (II). By Lemma 3.3, \ R0 \ L0 = Is, and thus S is a
subdirected product of S= \ R0 and S=L0 . Since S= \ R0 is a subdirect
product of S= and S=R0 , S= \ R0 2 SLG _ LBG = LBG. By Lemma 2.1,
S 2 O, hence S= \ R0 2 OLBG, S=L0 2 ORBG. Therefore S is a subdirect
product of a semigroup in OLBG and a semigroup in ORBG, as required.
(II) implies (I). This is obvious.
4. V \ (OLBG _ ORBG)
Theorem 4.1. Let V 2 LCR: Then
V \ (OLBG _ ORBG) = (V \ OLBG) _ (V \ ORBG):
Proof. Obviously V \ OLBG V \ (OLBG _ ORBG), V \ ORBG V \ (OLBG _ ORBG), and thus
(V \ OLBG) _ (V \ ORBG) V \ (OLBG _ ORBG):
Conversely, let S 2 V \ (OLBG _ ORBG) O. Then by Theorem 3.4,
S= 2 V \ SLG, S=R0 2 V \ ORBG. By Lemma 3.3, \ R0 \ L0 = Is, and
thus S is a subdirect product of S= ,S=L0 and S=R0 , hence S 2 (V \ SLG) _
(V \ OLBG) _ (V \ ORBG) = (V \ OLBG) _ (V \ ORBG). The required
conclusion now follows.
Theorem 4.2. Let V 2 LCR, and SLG V .Then
(V \ OLBG) _ (V \ ORBG) = O \ (V \ OLBG)Tr \ (V \ ORBG)T 1 :
Proof. By the implication (I))(II) of Theorem 2.3 and its dual, it is
easily checked that V \ OLBG, V \ ORBG O \ (V \ OLBG)Tr \ (V \
ORBG)T 1 , and thus (V \ OLBG) _ (V _ ORBG) O \ (V \ OLBG)Tr \ (V \
ORBG)T 1 .
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HAIXUAN YANG
Conversely, Let S 2 O \ (V \ OLBG)Tr \ (V \ ORBG)T 1 . Then by Lemma
2.1 and Notation 3.1, S= 2 SLG, S=R0 2 V \ OLBG, S=L0 2 V \ ORBG.
By Lemma 3.3, \ R0 \ L0 = Is, and thus S is a subdirect product of
S= , S=L0 and S=R0 , and thus S 2 (V \ OLBG) _ (V \ ORBG) _ SLG =
(V \ OLBG) _ (V \ ORBG) since SLG V , as required.
If SLG V OLBG _ ORBG, then by Theorem 4.1 and Theorem 4.2,
V = (V \ OLBG) _ (O \ ORBG) = O \ (V \ OLBG)Tr \ (V \ ORBG)T 1 .
And if V \ OLBG = u = v ] 2A , V \ ORBG = s = t ]2B , then by
Lemma 3.2, the identity of V could be given.
References
1] M. Petrich, Varietiesoforthodoxbandsofgroups, Pacic J. Math., 91(1980), 209-217.
2] T.E. Hall and P.R. Jones,Onthelatticeofvarietiesofbandsofgroups, 91(1980), 327-337.
3] N.R. Reilly, Varietiesofcompletelyregularsemigroups, J. Austral. Math. Soc. (series A)
38(1985), 372-393.
4] J.M. Howis, An introduction to semigroup theory, Academic Press, London/ New York,
1976.
5] Shuhua Zhang, CertainoperatorsrelatedtoMal'cevProductsonvarietiesofcompletelyregularsemigroups, J. Algebra 168(1994), 249-272.
6] M. Petrich and N.R. Reilly, Operatorsrelatedtoidempotentgeneratedandmonoidcompletelyregularsemigroups, J. Austral. Math . Soc., 49(1990), 1-29.
Department of Mathematics, Lanzhon University, Lanzhon 730000, China.