International Journal of Algebra, Vol. 2, 2008, no. 19, 925 - 931
The Maximum Idempotent Separating
Congruence on E-inversive E-semigroups
Manoj Siripitukdet
Department of Mathematics, Naresuan University
Phitsanulok 65000, Thailand
manojs@nu.ac.th
Supavinee Sattayaporn
Department of Mathematics, Uttaradit Rajabhat University
Uttaradit 53000, Thailand
supavinee.uru@windowslive.com
Abstract
A semigroup S is an E-inversive E-semigroup if for every a ∈ S,
there exists an element x ∈ S such that ax is idempotent and the
set of all idempotents of S forms a subsemigroup. The aim of this
paper is to investigate the maximum idempotent separating congruence
on E-inversive E-semigroups by using a full and weakly self-conjugate
subsemigroup.
Mathematics Subject Classification: 20M10
Keywords: E-inversive E-semigroup, maximum idempotent separating
congruence, full, weakly self-conjugate subsemigroup
1 Introduction and Preliminaries
Let S be a semigroup and E(S) denote the set of all idempotents of S.An
element a in a semigroup S is called E-inversive [1] if there exists x ∈ S such
that ax is idempotent of S. A semigroup S is called an E-inversive if every
element of S is E-inversive. A semigroup S is called an E-semigroup if E(S)
forms a subsemigroup of S. A semigroup S is said to be a band if every element
of S is idempotent, and a band S is rectangular if for all x, y ∈ S, x = xyx [3,
page 10]. For a semigroup S and a ∈ S, V (a):={x ∈ S | a = axa, x = xax}
is the set of all inverses of a and W (a):={x ∈ S | x = xax
} is the set of all
926 M. Siripitukdet and S. Sattayaporn
weak inverses of a. A congruence ρ on a semigroup S is called an idempotent
separating congruence if every ρ-class contains at most one idempotent.
In an E-inversive E-semigroup S, one important thing to note here is that
the maximum idempotent separating congruence on S in general may not exist,
see [1]. Weipoltshammer [1] described the idempotent separating congruence
on an E-inversive E-semigroup [1, Theorem 6.1] and described the maximum
idempotent separating congruence on an E-inversive semigroup such that E(S)
forms a rectangular band [1, Corollary 6.2]. Basic properties and results of E-
inversive E-semigroups were given by Mitsch [3] and Weipoltshammer [1].
In this paper, we investigated characterizations of the maximum idempo-
tent separating congruence on an E-inversive E-semigroup S by using a full
and weakly self-conjugate subsemigroups of S. The last theorem, we described
an idempotent separating congruence on S concerning the centralizer C
S
(H)
of H in S.
To present the main results we first recall some definitions and a relation
on a semigroup which is important here.
A subset H of a semigroup S is full [4] if E(S) ⊆ H. A subsemigroup H of
a semigroup S is called weakly self-conjugate if for all a ∈ S, x ∈ H, a
∈ W (a),
we have axa
,a
xa ∈ H. For any subsets H and B of a semigroup S, let
H
ω
B
:= {a ∈ S | ba ∈ H for some b ∈ B}.
If B = H then H
ω
H
will b e denoted by H
ω
and it is called the closure of H.
If H is a subsemigroup of a semigroup S, then H ⊆ H
ω
. H is called a closed
subsemigroup [4] of S if H = H
ω
.
For any nonempty subset H of a semigroup S, we define a relation δ on S
as follows :
δ := {(a, b) ∈ S × S | for all a
∈ W (a) there exists b
∈ W (b)
such that axa
= bxb
,a
xa = b
xb for all x ∈ H, and for
all b
∈ W (b) there exists a
∈ W (a) such that axa
= bxb
,
a
xa = b
xb for all x ∈ H}.
Note that δ may be an empty set. If S is an E-inversive semigroup, then
(a, a) ∈ δ for all a ∈ S,soδ is not an empty set.
For basic concepts in semigroup theory, see[2] and [5] and for examples of E-
inversive E-semigroups, see [1].
The following results are used in this research.
Lemma 1.1. [1] A semigroup S is E-inversive if and only if W (a) = ∅ for
all a ∈ S.
Maximum idempotent separating congruence 927
Proposition 1.2. [1] For any semigroup S, the following statements are
equivalent:
(i) S is an E-semigroup.
(ii) W (ab)=W (b)W (a) for all a, b ∈ S.
Proposition 1.3. [1] Let S be an E-semigroup. Then
(i) for all a ∈ S, a
∈ W (a),e,f ∈ E(S),ea
,a
f, fa
e ∈ W (a),
(ii) for all a ∈ S, a
∈ W (a),e∈ E(S),a
ea, aea
∈ E(S),
(iii) for all e ∈ E(S),W(e) ⊆ E(S),
(iv) for all e, f ∈ E(S),W(ef)=W (fe).
Proposition 1.4. [1] For any E- inversive semigroup S, the following are
equivalent.
(i) E(S) is a rectangular band.
(ii) For al l a, b ∈ S, W (a) ∩ W (b) = ∅ implies W (a)=W (b).
Proposition 1.5. Let S be an E-semigroup. If a
∈ V (a) for all a ∈ S then
W (a)=W (a
a)a
W (aa
).
Proof. Let a ∈ S and a
∈ V (a). Let x ∈ W (a
a) and y ∈ W (aa
). By
Proposition 1.3(iii) and (i), we have W (a
a) ⊆ E(S), W (aa
) ⊆ E(S) and
xa
y ∈ W (a), respectively. So W (a
a)a
W (aa
) ⊆ W (a).
Let z ∈ W (a) and a
∈ V (a),a = aa
a. Consider z = zaz = z(aa
a)z =
(za)a
(az) and (za)(a
a)(za)=z(aa
a)za = zaza = za. Then za ∈ W (a
a).
Similarly, we have az =(az)(aa
)az and so az ∈ W (aa
). Therefore z ∈
W (a
a)a
W (aa
), so W (a) ⊆ W (a
a)a
W (aa
) and W (a)=W (a
a)a
W (aa
).
A subset H of a semigroup S is called unitary if for all a ∈ S, and for all
h ∈ H, ha ∈ Hor ah ∈ H implies a ∈ H.
We have the following properties :
Proposition 1.6. Let S be an E-inversive semigroup with a full subset H
of S. Then H is unitary if and only if H is closed.
Proof. Suppose that H is unitary. Let x ∈ H
ω
. Then there exists h ∈ H
such that hx ∈ H, which implies that x ∈ H, and so H
ω
⊆ H. Since H is a
subsemigroup of S, we have H ⊆ H
ω
. Hence H = H
ω
.
Conversely, let hx, h ∈ H. Then x ∈ H
ω
. Since H is closed, we have x ∈ H.
If h, xh ∈ H and x
∈ W (x), then (x
xh)x ∈ H since H is full. It follows that
x ∈ H
ω
= H. Then H is unitary.
Proposition 1.7. Every full and closed subsemigroup of an E-inverse semi-
group is E-inversive.
928 M. Siripitukdet and S. Sattayaporn
Proof. Let H be a full and closed subsemigroup of an E-inversive semigroup
S. Let h ∈ H and h
∈ W (h). Then hh
∈ E(S) ⊆ H,soh
∈ H
ω
. Since H is
closed, h
∈ H. This shows that H is an E-inversive subsemigroup of S.
By Propositions 1.6 and 1.7, we have
Proposition 1.8. Every full and unitary subsemigroup of an E-inversive
semigroup contains all the weak inverses of its elements.
Proof. Let H be a full and unitary subsemigroup of an E-inversive semigroup
S. We shall show that for all a ∈ H, a
∈ W (a) implies W (a) ⊆ H. Let a ∈ H
and a
∈ W (a). Then a
a, aa
∈ E(S) ⊆ H. Since H is unitary, it follows that
a
∈ H and W (a) ⊆ H. This shows that H contains all the weak inverses of
its elements.
Proposition 1.9. Let S be an E-inversive semigroup. If H is a full and
closed subsemigroup of S, then E ⊆ H
ω
E
= H
ω
where E = E(S).
Proof. Clearly, E(S) ⊆ H
ω
E
. Let x ∈ H
ω
E
. Then there exists e ∈ E(S)
such that ex ∈ H. Since H is full, we have e ∈ H and so x ∈ H
ω
. Therefore
H
ω
E
⊆ H
ω
. Let y ∈ H
ω
. By Proposition 1.7, there exists h
∈ W (h) ∩ H such
that (h
h)y = h
(hy) ∈ H. Since h
h ∈ E(S), we have y ∈ H
ω
E
. It follows that
H
ω
E
= H
ω
.
Hence the proof is completed.
2 Main Results
The idempotent separating congruence μ on an E-inversive E-semigroups can
be found by Weipoltshammer [1] as follows :
μ := {(a, b) ∈ S × S | for all a
∈ W (a) there exists
b
∈ W (b), aea
= beb
,a
ea = b
eb for all e ∈ E(S)
and for all b
∈ W (b) there exists a
∈ W (a),
aea
= beb
,a
ea = b
eb for all e ∈ E(S)}. (∗)
Let C be the class of all full and weakly self-conjugate subsemigroups of a
semigroup S.ForH ∈C, we replace E(S)in(∗)byH. Then we have
Theorem 2.1. If S is an E-inversive E-semigroup and H ∈C, then a
binary relation
δ := {(a, b) ∈ S × S | f or each a
∈ W (a) there exists
b
∈ W (b),a
xa = b
xb, axa
= bxb
for all x ∈ H
and for all b
∈ W (b) there exists a
∈ W (a),
a
xa = b
xb, axa
= bxb
for all x ∈ H}
Maximum idempotent separating congruence 929
is an idempotent separating congruence on S. Moreover, if E(S) is a rectan-
gular band with E(S)=H
ω
E
, then δ is the maximum idempotent separating
congruence on S.
Proof. Obviously, δ is reflexive and symmetric. Let a, b, c be elements in S
such that aδb and bδc and let a
∈ W (a). Then there exists b
∈ W (b) such
that a
xa = b
xb, axa
= bxb
for all x ∈ H. Since bδc and b
∈ W (b), there is
c
∈ W (c) such that b
xb = c
xc, bxb
= cxc
for all x ∈ H.Thusa
xa = c
xc
and axa
= cxc
for all x ∈ H.
Similarly, we can show that for all c
∈ W (c), there exists a
∈ W (a) such
that a
xa = c
xc and axa
= cxc
for all x ∈ H. Hence aδc.
Let a, b, c ∈ S with aδb, and let a
∈ W (a). Then there exists b
∈ W (b)
such that a
xa = b
xb, axa
= bxb
for all x ∈ H. Let c
∈ W (c). By c
a
∈
W (ac) and c
b
∈ W (bc), we get that for all x ∈ H,
(ac)x(c
a
)=a(cxc
)a
= b(cxc
)b
=(bc)x(c
b
) since cxc
∈ H,
and
(c
a
)x(ac)=c
(a
xa)c
= c
(b
xb)c
=(c
b
)x(bc) since b
xb ∈ H.
Hence acδbc,soδ is a right compatible. Similarly, we can show that δ is a left
compatible. Therefore δ is a congruence on S.
Let e, f be elements in S such that eδf . Since e ∈ W (e), there exists
f
∈ W (f) such that exe = f
xf = fxf
for all x ∈ H.
Since H is full, e ∈ H and e = eee = f
ef = fef
, we have ef =(f
ef)f =
f
ef = e.Nowf ∈ W (f). There exists e
∈ W (e) such that fxf = e
xe = exe
for all x ∈ H. Since f ∈ H, f = fff = e
fe = efe
, it follows that ef =
e(efe
)=efe
= f . Therefore e = f and so δ is an idempotent separating
congruence on S.
Suppose that E(S) is a rectangular band with E(S)=H
ω
E
. Let ρ be
an arbitrary idempotent separating congruence on S. Let a, b ∈ S with aρb,
and let a
∈ W (a). We choose b
∗
∈ W (b). Then b
∗
aρb
∗
b. Let b
= a
ab
∗
aa
.
By Proposition 1.3(i), we have b
∈ W (b). For any x ∈ H, a
xa ρ a
xb =
(a
aa
)xb. Since a
a, b
∗
b ∈ E(S) and E(S) is a rectangular band, it follows
that a
a =(a
a)(b
∗
b)(a
a). Thus a
xaρ(a
ab
∗
ba
a)a
xb. Since b
∗
aρb
∗
b, we have
a
xaρa
a(b
∗
a)a
aa
xb = a
ab
∗
aa
xb = b
xb.Thusa
xaρb
xb.Nowa
xa, b
xb ∈ H
because H is weakly self-conjugate. Since E(S)=H
ω
E
and a
xa = a
a(a
xa) ∈
H, we have a
xa ∈ H
ω
E
= E(S). Similarly, b
xb = b
b(b
xb) ∈ H.Thus
b
xb ∈ H
ω
E
= E(S). Since ρ is an idempotent separating congruence on S,we