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**On Primitive Abundant Semigroups and
PA-Blocked Rees Matrix Semigroups**

Lawrence N. EZEAKO

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*Department of
Mathematics and Computer Science, Federal Univesity of Technoly, Minna, Nigeria*

**Abstract**

In this paper we utilize Warne’s homomorphism theorem for bisimple inverse semigroups with an identity element [13, Theorem 1.1] to prove the theorem: Every primitive abundant semigroup with a zero element is isormorphic to a PA-blocked Ree’s matrix semigroup.

Our approach obviates the tedium of the proof of the existence of a homomorphic mapping (see Markie, 1975) in order to prove the same theorem.

**Keywords**

Bisimple; Inverse; Semigroups; Primitive Abundant Semigroups; Ree’s Matrix Semigroups; Homomorphism

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**Preliminaries**

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*Ree’s matrix Semigroup (A completely O-Simple Semigroup)*

Let G º a simple group with
identity e _{}

P º matrix (pm with entries from the zero group G^{0},
where

_{}

G^{0} º (GỤ{0} i.e.
a L × I matrix

Let S = (GxIxL)u {0}

where

Then S thus constructed is completely O - simple, and S º M^{o} [G,I,L,P] is the I × L, Ree’s
matrix semigroup over the O-group G^{o}, with the regular sandwich
matrix P.

*The PA - blocked Ree’s Matrix Semigroup *

D. Rees [9] has shown that we can construct a semigroup from a set of monoids and bisystems over these monoids. Such a construction generalizes Rees matrix semigroup. Under certain conditions (see [9]) such as semi group becomes abundant, with all its non-zero idempotents primitive. It is called a PA-blocked Rees matrix semmigroup.

*Important remarks: Since S is O-Simple*

· {0} and S are the only ideals of S

· S x S ¹ 0

· "aÎ s/ {0}; aS = Sa = S

Also, since S is
completely 0 - simple, S satisfies the Min_{L}, Min_{R}
conditions i.e every non-empty set of the l - classes or the R-classes
possesses a minimal member

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*Right [Left] S- Systems*

If M is a set and S is a monoid then M is a right S-system if there exists a mapping (x,s) ® xs from M x S into M, with the properties that

i. _{}

ii. _{}

a left S -
system is dually defined. If S and T are monoids, then M is an (St) - Bisystem,
where M is a left S - system as well as a right T - system and for each _{}

we have _{}

If M is a right S - system and N is a left S - system and _{} is an
equivalence relation on M** **_{}N generated by the subset ** _{} **of

* using these definition we make the following propositions

**Propositions**

Let e, f be idempotent of a primitive abundant semigroup S, with zero, if e, f are not D-related and if ef, fe are both non-zero, the;

1. H* ef and H* fe are cancellative subsemigroups of S without identities

2. /H*e/ = /H * ef/ = /H* fe/ = /H* f/

3. H* ef is isomorphic as a semigroup and as a right H* f - system to a left idea of H*f, and isomorphic as a semigroup and as a left H*e - system to a left ideal of H*e

*Proof of Proposition 1*

_{}

_{}

_{}

_{}

_{}_{}

_{}

*Proof of Proposition 2*

_{}

_{}

Since _{}

_{} (a)

Also, if x Î H^{*} f,
then Fx Î H^{*} f and ex = x

Thus

If x_{1}, x_{2} Î
H^{*} f and fx_{1} = fs_{2}, then

_{}

**_{}** (b)

From (a) and (b) Þ /H^{*}f/
= /H^{*} ef/

Similarly, /H^{*} ef/ = H^{*}el

and 2 is proved.

*Proof of Proposition 3*

Let _{}

Also, _{}

For each* _{}*

_{}

Furthermore, for any* _{}*

_{}

So, F is a semi group isomorphism

Similarly, we can show that H^{*} ef is isomorphic as

a left H^{*} e - system to a left ideal H^{*}e

i.e. 3 is proved*.*

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** Construction
of a Primitive Abundant Semigroup S, with Zero**

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_{}

_{}

_{}

_{}

** Main
Theorem**

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* *Every
primitive Abundant Semigroup as constructed in (3.0) above is Isomorphic to A
PA-Block Rrees Matrix Semigroup

** Proof
of Main Theorem**

_{}

_{}

_{}

** **

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**Conclusion**

We have thus proved in a very simple manner i.e. without recourse to the necessity for the existence of a homomorphic mapping,-that every primitive abundant semigroup with a zero, is isomorphic to a PA - blocked Rees Matrix semigroup

**Acknowledgements**

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

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**References**

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1.
Armstrong S., *The Structure of Type A Semigroups*, Semigroup Forum
1984, 29, 319-336, Springer Verlag, New York, Inc.

2.
Clifford H., Preston G. B., *The Algebraic Theory of Semigroups*, Math.
Surveys, 1961, 7(1) & 1967, II Amer. Math. Soc. Providence, R.I.

3.
Fountain J. B., *Adequate Semigroups*, Proc. Edin. Math. Soc., 1979,
22, p. 113-125

4.
Fountain J. B., *Adequate Semigroups*, Proc. London Math. Soc., 1982,
44(3), p. 103-129

5.
Howie J. M., *An Introduction To Semigroup Theory*, Academic Press,
1976.

6.
Markie L., *On Locally Regular Rees Matrix Semigroups*, Acta. Sci.
Math (Szeged), 1975, 37, p. 95-102.

7.
Munn W. D., *A Class of Irreducible Matrix Representation of an
Arbitrary Inverse Semigroup*, Proc. Glasgow Math. Assoc., 1961, 5, p. 41-48.

8.
Pastigin F., *A Representation of a Semigroup of Matrices Over a Group
with Zero*, Semigroup Forum, 1975, 10, p. 238-249.

9.
Rees D., *On Semigroups*, Proc. Cambridge Philos. Soc., 1940, 36, p.
387-400.

10. Rees
D., *On The Ideal Structure of a Semigroup Satisfying a Cancellation Law*,
Quant. J. Math. Oxford Ser., 1948, 19(2), p. 101-108.

11. Reilly
M. R., Clifford A. H., *Bisimple Inverse Semigroup as Semigroups of Ordered
Triples*, C and .J. Math., 1968, 20, p. 25-39.

12. Steinfeld
O., *On a Generalization of Completely O-Simple Semigroups*, Acta. Sci.
Math. (Szeged), 1967, 28, p. 35-46.

13. Warne
R. J., *Homomorphisms of **a**
- Simple Inverse Semigroups with Identity Pacific*, J. Math., 1964, 14, p. 1111-1122.