Advances in Pure Mathematics, 2011, 1, 73-76
doi:10.4236/apm.2011.13016 Published Online May 2011 (http://www.scirp.org/journal/apm)
Copyright © 2011 SciRes. APM
A Study on the Conversion of a Sem igroup
to a Semilattice
Bahman Tabatabaie, Seyed Mostafa Zebarjad
Department of Mat hematics, Shiraz University, Shiraz, Iran
E-mail: s_zebarjad@yahoo.com
Received January 19, 2011; revised March 15, 2011; accepted March 25, 2011
Abstract
The main aim of the current research has been concentrated to clarify the condition for converting the inverse
semigroups such as S to a semilattice. For this purpose a property the so-called *unitaryE has been de-
fined and it has been tried to prove that each inverse semigroups limited with *unitaryE show the specifi-
cation of a semilattice.
Keywords: Semigroup, Semilattice, *unitaryE
1. Introduction
1.1. Literature Survey
Literature survey done by th e authors show that a special
class of semigroups possessing is formed by the *
E
unitary inverse semigroups, sometimes also called 0
*unitaryE, which was defined by Szendrei [1] and has
been intensely studied in the semigroup literature. See,
for example, Kellendonk’s topological groupoid is
Hausdorff when S is *unitaryE[2], and the related
class of unitaryE inverse semigroups have also been
shown to prov ide Hausdorf f groupoids [3 ]. In the current
research the authors try to prove that each inverse semi-
groups limited with *unitaryE show the specification
of a semilattice. For this purpose, firstly we present ele-
mentary concepts as follows.
1.2. Preliminary Definitions and Propositions
A groupoid is a set G together with a subs et 2
GGG
,
a product map

,ab ab.
From 2
G to G, and an inverse map 1
aa
(so
that

1
1
aa
) from G onto G such that:
1) if

2
,,, ,ab bcG then

2
,,,ab ca bcG
and
 
ab cabc.
2)

12
,bb G
for all bG
, and if

2
,ab G
then

1
aabb
and

1.ab ba
Note that 2
G is nothing but the set of all pairs

,
x
y in GG for which
x
y is defined, and 2
G is
called the set of composable pairs of the groupoid
G[3].
If
1
,
Gd xxx
 is the domain of
x
and
1
rx xx
is its range. The pair

,
x
y is composable
if and only if the range of y is the domain of
x
.
0
GdGrG is the unit space of G, its elements
are units in sense that
d
x
xx and

rx x [4].
By an inverse semigroup we mean a semigroup S
such that for each a in S, there exists a unique ele-
ment a
in S with the following properties:
,aa aa
and aaa a
 
It is well known that the correspondence aa
is
an involutive anti-homomorphism, i.e.,

abb a
for all a and b in S. It is very common to denote it
by
ES, the set of all idempotent elements of S, it
means that 2
aa
for all a in

ES. It is clear that
aa
for all a in
ES.
A very important example of an inverse semigroup is
given by
SIX the set of all partial one-to-one
maps on a set
X
. So each element of

I
X is a bijec-
tion form a subset U of
X
onto another subset V
of
X
. The set
I
X is a semigroup where the multi-
plication rule is given by composition of partial maps
with the largest possible domain.
For example, if

12
,
I
X

with 11 1
:UV
and
22 2
:UV
, then
 
1
122211 21
:VU VU
 
 
is given by:


121 2.aa
 
B. TABATABAIE ET AL.
Copyright © 2011 SciRes. APM
74
The element 1
is taken to be 1
1
. It is easily checked
that

I
X is an inverse semigroup [3,5].
We recall that a relation
on a set
X
is called a
partial ordering of
X
if for all ,,abc X:
1) aa
2) ab and ba implies ab
3) ab and bc implies ac.
The following example is of great importance to us.
Define


,efefES to mean .effe e It is
clear that is a partial ordering of

ES. We shall
call the natural partial ordering of

ES.
An element b of a partially ordered set
X
is called
an upper bound of a subset Y of
X
, if yb
for
each y in Y. An upper bound b of Y is called a
least upper bound or join of Y, if bc for every up-
per bound c of Y. If Y has a join in
X
, it is clearly
unique. Lower bound and greatest lower bound or meet
can be defined sim i l a rly.
A partially ordered set
X
is called a semilattice if
every two elements subset

,ab of
X
has a join and
a meet in
X
; it implies that every finite subset of
X
has both a join and a meet. The join (or meet) of
,ab
will be denoted by ab (or ab)[3].
Definition 1.1 Suppose that S is an inverse semigroup
and
X
can be assumed that as a locally compact
Hausdorff topological space.
An action of S on
X
is a semigroup homomorph-
ism as follows:
:SIX
a
a
such that
1) for every aS there is a continuous a
with
open dom a i n in
X
.
2) the union of the domains of all the a
coincides
with
X
.
Proposition 1.2 Let S be an inverse semigroup,
an action of S on a set
X
and aS, then
and
aaa a
aaaa
 


Proof: Since
is an action of S on
X
then

:SIX
is a semigroup homomorphism, so for
every aS we have



aa aa
 
, then
aaa
a
 
, and simillary a
aaa
 
 
With regard to the above text one may conclude that,
1
a
a
, and if

eES, so e
is the iden tity map
on its domain.
Since the range of each a
coincides with the do-
main of 1
a
a
, therefore it can be open as well as its
domain. Also it can be mentioned that 1
a
, is continu-
ous, so a
is necessarily a homeomorphism onto its
range.
For every
eES the domain (and range) of e
can be denoted by e
E, it means:
:.
ee e
EE
It is clear to show that the domains of both a
and
aa
is the same, and implies that the domain of a
is
aa
E
. Likewise the range of a
is given by aa
E
. Thus
:
aaa aa
EE
is a homeomorphism for every aS
.
Briefly if e and
f
are in

ES then we have
ef ef

and efef
EEE
.
Proposition 1.3 For each aS and
eES we
have:
ae aa aea
EE E

Proof: Since N. Sieben [6], R. Ex el [7] and Lawson [8 ]
proved it, the author s use their result.
Definition 1.4 Let
be the subset of SX
given
by:

:aa
abSXbE

and for every
11
,ab and

22
,ab in we will say
that
112 2
,~,abab if 12
bb and there exists an
idempotent e in
ES such that 1,
e
bE
and
12
ae ae
.
It is clearly that the relation ~ is an equivalence re-
lation on
The equivalence class of

,ab will be
denoted by
,ab .
Let
,: ,GabaSbX and put
 

2
1122
212
,,,:a
ab abGGGbb

And for every
2
112 2
,,,ab abG define:


1
112 212 2
1
111 1
,, ,
,,
a
ab abaab
aba b

it is easy to see that G is a groupoid [3] and the unit
space

0
G of G naturally identifies with
X
under
the correspondence


0
,,eGbbX
where e is any idempotent such that S
eE. We show
G semigroup as
,,GSX
.
We would now like to give G is a topology. Let
aS
and U be an open subset of aa
E we define
,aU
as follows:
,,:aUabGbU

The collection of all
,aU
is the basis of a topol-
ogy on G, and also the multiplication and inversion
operations on G are continuous, therefore G is a to-
pological grou poid.
B. TABATABAIE ET AL.
Copyright © 2011 SciRes. APM
75
2. Main Results
Recall from [2] that an inverse semigroup S is natu-
rally equipped with a partial order defined by:
ab abaaaS

Proposition 2.1 Assume that S is an inverse semi-
group which is a semilattice. Suppose that
is an ac-
tion of S on a locally compact Hausdorff space
X
,
such that for each aS, the domain aa
Eof a
is
closed. Then
,,GG SX
is Hausdorff.
Proof:Suppose
,ac and
,bd are two distinct
elements of

,,GSX
. The aim is to find two disjoint
open subsets 1
T and 2
T of

,,GSX
such that:
1212
,,, ,acTbdT TT

We consider two cases:
Case 1): If

cd:
Since
X
is Hausdorff space then

12121 2
,open,,,FFXc FdFFF
 
Now let

11
,aa
TaFE
 and

22
,bb
TbFE

Since 1
T and 2
T are open set and


11
,: ,
aa
TakGkFE



22
,: ,
bb
TbkGkFE

It is clearly that:
1212
,,, andacTbdTTT
 
Case 2): If

cd:
Since S is a semilattice let hab so

,,
ha hahhac bc
hb hbhh



Then referring to what proposed in Definition 1.4.
hh
cE
. But hh
E is closed then 2\hh
TXE
can be
open and 2
cT.
Now we can set T as 2aa bb
TE E

. But we
know that


,:,kTa Tak
 and it is clear that
,,,,,acaT bcbT

.
To do so it is enough to prove that
,aT

,bT
.
Suppose that
,, ,lk aTbT

 then:





 
 

,,,,,~,
,,
,,,,,~,
,,
e
f
lkaTlk aklkak
eESk Eaele
lkbTlk bklkbk
fESkEbflf

 

 
Since

efE S and ef fe,

eef
kE E , it can
be replaced e and f with ef and finally we have:
,aeflefleflfebfebef
 
Therefore we can find an element
eES such
that ,,.
e
kEae lele be
 So

leleael le
lellellleele

, then lea, and similary le b
,
since hab
thus le h
, then leleh h
, hence
lle llehh hh

, and finally
e
lllle hh
kEE EE


But kT
that is contradicts.
Definition 2.2 A zero in an inverse semigroup S is
an element 0S
such that:
00oa aa S

Definition 2.3 An inverse semigroup S with zero is
said to be unitaryE if for every ,ea S one has
that 22
eeaaa
 .
In other words, if an element dominates a nonzero
idempotent then that element itself is an idempotent.
Proposition 2.4 If S is a unitaryE inverse se-
migroup and ,ab belong to the defined semigroup S
such that aa bb
and ae be for some nonzero
idempotent eaa
then .
ab
Proof: We define
x
aea
. So
x
is nonzero
idempotent because:

*
e aaeaaaae eaaaa
 

Then eaaeaa
(because of the ability of idempo-
tent elements for being commute) and we have
.baxba aeabb beabeaaeax


Therefore, we have
x
ba
. Since S is a E
unitary which implies that ba is idempotent. Then

ba baab


so ab
is idempotent as well.
But, we have
bbbb bbbaabab baaa aaaa
   

Setting ,
y
ba b
we have that
y
yb ab ba bb aaaa bb aa bbbb bbb
    

Also
y
yaa
, while
,and ,bbbbby yaaaaay y
 

So it is enough to prove that
y
ay

. We have
ayabababbabb babb abby
 

In what follows we give the main result of this paper.
Theorem 2.5 In condition that S is a unitaryE
inverse semigroup with zero, thencan be appeared as a
semilattice.
Proof: For proving the above theorem it is necessary to
show that ab
exists for every ,ab S. If there is not
nonzero hS
such that ,hab, it is obvious that
B. TABATABAIE ET AL.
Copyright © 2011 SciRes. APM
76
0
ab and it can be satisfied for the proof.
For doing this we can assume that there is a nonzero
hS in which ,hab. Our claim is that .ab bba a

Suppose that kaabb

and considering to our as-
sumption

,,hh aabb

we have hh k
.
Substituting
x
ak and ybk,
2
2
xx kaakkk
x
xyy
yy kbbkkk
 




also
x
h hakh hah hhbh hbkh hyh h
 

Using the proposition

2.4
x
y
will be achieved and
so

ab baa ab bakxybk
baabbbbbaabaa

 

 
and finally
ab bba a

(1)
By applying the above argument to a,b,h
and
knowing that 0h and ,hab

we have
abb baa
 
so

**
abb baa
 
and therefore Equation (1) can be modified to
2:
bb aaa b

(2)
We have that ,hab then hahh
and hbhh
,
then we can show that
bahh bbhh hh
 

Since S is a unitaryE and ba
is dominated by
hh
, we have

2
baba

. By applying the same rea-
soning to ,ab

and h,

2
ba ba
can be a result.
Thus


*
baba
ba ba


and hence abb babbba


ab bbb a
(3)
By combination of Equations (1) to (3), Equation (4)
will be appeared.
ab bba abbaaa b

 (4)
At the end we try to prove that ab b
can satisfy the
following condition
,habbab

for every hS
such that ,hab.
It is clear that ,ab bab
and as defined before
kaabb
, then we have *
hh k, and so

hahhakh haaab bh hab bhhab bhh
 
 
Finally habb
. It means that abb
is the join of
a and b and this is the proof of theorem.
3. References
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Theorem for E-Unitary Regular Semigroups,” Acta Sci-
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[2] M. V. Lawson, “Inverse Semigroups: The Theory of Par-
tial Symmetries,” Word Scientific, Si ngapore, 1998.
doi:10.1142/9789812816689
[3] J. Renault, “A Groupoid Approach to C*-Algebra, Lec-
ture Notes in Mathematics,” 1st Edition, Sprin-
ger-Verlang, Berlin, Vol. 793, 1980.
[4] J. M. Howie, “Fundamentals of Semigroup Theory,”
Clarendon Press, Oxford, 1995.
[5] H. Clifford and G. B. Preston, “The Algebric Theory of
Semigroups,” American Mathematical Society, United
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[6] N. Sieben, “C*-Crossed Products by Partial Actions and
Actions of Inverse Semigroups,” Journal of the Austra-
lian Mathematical Society, Series A, Vol. 63, No. 1, 1997,
pp. 32-46.
[7] R. Exel, “Inverse Semigroups and Combinatorial C*-
Algebras,” Bulletin of the Brazilian Mathematical Society,
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[8] M. V. Lawson, “Inverse Semigroups, The Theory of Par-
tial Symmetries,” Word Scientific, Singapore, 1998.