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

abochoche@gmail.com

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 = { e₀, e₁, e₂,………………} with  e₀> e₁> e₂

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 pⁿ, m and n are nonnegative integers. We also let pq = p⁰ = q⁰ = 1. Then each element q^m p^m Î  B is unique. It is trivial to show that B is indeed a semigroup under the composition, multiplication.

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  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  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 = qⁿpⁿ (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^maⁿ where m,n are nonnegative integers.

And a⁰ = b⁰ = e⁰. 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 aⁿ ¹ a^r for 1 £r<n

For all nonnegative integer n.

(which is a contradiction). So, a² ¹ a.

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

Let r be such that 1 ≤ r ≤ n + 1 and a_n+1 = a^r

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

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

So, S is infinite.

Also, aⁿb^mÞa^maⁿ = a^mbm Þ a^m+n = e = a⁰ 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 aⁿ ¹ b^m unless m = n = 0.

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

Thus we have;

Hence the expression bⁿa^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 e₁ Î S, say, such that

Þ e = e₁ (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.