Advances in Pure Mathematics, 2011, 1, 73-76
doi:10.4236/apm.2011.13016 Published Online May 2011 (http://www.scirp.org/journal/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
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
,
x
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. 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. 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
Eof 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. 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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[2] M. V. Lawson, “Inverse Semigroups: The Theory of Par-
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