**On an Inverse Semi-Group which is
Simple but Not Completely Simple**

Lawrence N. EZEAKO

* *

*Department of
Mathematics and Computer Science, Federal University of Technology, Minna,
Nigeria*

** **

** **

**Abstract**

W. D. Munn, (1966) has shown that if E is a semilattice, then the Munn semigroup
T_{E} of E is an inverse semigroup whose semilattice of idempotents is
isomorphic to E. If E is a uniform semillatice, then T_{E} is a bisimple
inverse semigroup. J. M. Howie, (1976) proved that up to isomorphism, the only
fundamental bisimple semigroup S, having

E = C_{W} = { e_{0}, e_{1}, e_{2},………………}
with e_{0}> e_{1}> e_{2}

Is the Bisyclic semigroup. In this paper is highlighted the structure of the Bicyclic semigroup and prove that every simple semigroup which has idempotent, none of which is primitive, must necessarily be a web of bicyclic semigroups.

**Keywords**

Bicycle Semi-Group Structure; Inverse Semi-Group; Simple Semi-group; Bisimple Semi-Group, Completely Simple Semi-group

**Preliminaries**

** Construction**

Let X be a 2- element set, i.e. X = { p,q}. We construct a semigroup B,
from X by letting all elements in B, be of the form *q ^{m} p^{n}*,
m and n are nonnegative integers. We also let

*Definitions*

1
Let *a, b **Î S*,
where *S* is a semigroup. Then a and b are (generalized) inverse of each
other if *aba = a and bab = b*. S is called an inverse semigroup if every
element of S has a unique inverse in S.

2
A semigroup *S*, is __simple__ if S has no proper ideals

3
For every element *a, b* is a semi group *S*,

_{}

4 A semigroup S, is bisimple (i.e. D – simple) if S consist of a simple D-class.

5 A completely simple semigroup is a semigroup which contains a primitive idempotent.

*Propositions*

1. The semigroup B, constructed in section 1.1 above is the Bicyclic Semigroup. It is an inverse semigroup which is bisimple (hence simple), with an identity element. The idempotent of B are of the form.

_{}

And so B has no primitive idempotent i.e B is not completely simple (Prop. 1)

2. Every simple semigroup which has idempotents, none of which is primitive, must contain B. Such a seigroup has not be a “web of B’ s” (Prop. 2)

*Proof of Propositions*

*Proof of Prop 1*

1. B is an inverse Semigroup

_{}

Hence B is an inverse semigroup.

2. B is a Bisimle Semigroup

Let
a _{}

We
now show that a Â b [a *L* b]

i.e. aB = bB = B [Ba = Bb = B]

_{}

We cannot compare aB and bB under these conditions. So we let _{} and let min
[m,n] = (m,n) and define

_{}

Then let _{} which is possible

_{}

i.e. a Â b

Conversely; suppose that [m,n] Â [r,s], then there exist

_{}

_{}

** **

** **

**Existence and Nature of Idempotents in B**

_{}

Thus the idempotents of B are of the form e_{n} = q^{n}p^{n}
(n = 0,1,2,3,……….)

_{}

Thus B has no primitive idempotent

Þ B is a Bisimple semigroup Þ B is a simple semigroup

Þ B is not completely simple.

**The H – classes of B and the Left [Right] ideals of B**

_{}

Hence the H-class of B are the singletons _{}

**Left Ideals: **Let L be a left ideal of B. Then L = ÈB b, b ÈÎL

_{}

** **

*Proof of Prop. 2*

*Theorem 1*

Let e be a non – primitive idempotent of a simple semigroup S, which is not completely simple. Then S contains a bicyclic subsemigroup which has e as their identity element.

** **

*Lemma 1*

Let *e, Ia, Ib*, be elements of a semigroup S, such that; ea = ae =
a; eb = be = b, ab =e

But ba ¹ e. Then every element of the subsemigroup
generated by a and b is uniquely expressible as b^{m}a^{n}
where m,n are nonnegative integers.

And a^{0}
= b^{0} = e^{0}. Hence the subsemigroup of S generated by a and
b is isomorphic to the bicyclic semigroup B

*Proof of Lemma I*

__S is infinite__**:** We prove that a^{n} ¹ a^{r} for 1 £r<n

For all nonnegative integer n.

_{}

(which is a
contradiction). So, a^{2} ¹ a.

Suppose
that a, is such the a^{n} ≠ a^{k}, for 1 ≤ k <
n.

Let r be such that 1 ≤ r ≤ n + 1 and a_{n+1} = a^{r}

Then a^{n}(ab) = a^{r-1}(ab) Þ a^{n}e = a^{r-1}e Þ an = ar – 1, but 1 ≤ r ≤ n-1 Þ 1 ≤ r -1 < n, hence a^{n+1}
≠ a^{r} for 1 ≤ r ≤ n+1.

We conclude that {a^{n}; n = 0, 1,2,3,………..} is infinite.

So, S is infinite.

Also, a^{n}b^{m}Þa^{m}a^{n}
= a^{mb}m Þ a^{m+n} = e
= a^{0}^{ }by the above reasoning. i.e. m + n = 0. But both m
and n are nonnegative integer so we must have m = n = 0.

Thus a^{n} ¹ b^{m}
unless m = n = 0.

Now, suppose that; b^{n}a^{m} = b^{r}a^{s},
we may assume that n £ r, without loss
of generality.

Thus we have;

_{}

Hence the expression* b ^{n}a^{m}* is unique.

Therefore the subsemigroup generated by {a,b} is isomorphic to the bicyclic semigroup B.

*Proof of Theorem I*

Let e Î S be a non–primitive
idempotent. Then there exits an idempotent e_{1} Î S, say, such that

_{}

_{}

Þ e = e_{1} (which is a
contradiction!!) since e ¹ e.

So ba ¹ e. Hence the conditions in Lemma I are satisfied. So the bisyclic subsemigroup generated by {a,b} is a subsemigroup of S containing e as an identity element.

**Conclusion**

From
theorem I and the structure of the bicyclic semigroup *B*, thus discussed,
it is evident that any semigroup *S*, which is simple, and which contains
a non-primitive idempotent which acts as an identity element, must comprise *a*
“*web of B’*s.”

**Acknowledgements**

I sincerely wish to thank Professor K. R. Adeboye, (former Dean, School of Science Education, Federal University of Technology, Minna) for his very useful comments on an earlier draft of this paper.

**References**

** **

1.
Howie J. M., *An Introduction To Semigroup Theory*, Academic Press,
London, 1976.

2.
Munn W. D., *Uniform Semilattices and Bisimple. Inverse Semigroup*,
Quart .J. Math, Oxford, 1966, 17, p. 151-159.

3.
Reilly N. R., *Bisimple w-semigroups*, Proc. Glasgow Math. Assoc., 1966,
7, p. 160-167.

4.
Warne R. J., *I-bisimple Semigroups*, Traner. Amer. Math. Soc., 1968,
130, p. 367-386.