On Primitive Abundant Semigroups and PA-Blocked Rees Matrix Semigroups


Lawrence N. EZEAKO


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





In this paper we utilize Warnes 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 Rees 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.


Bisimple; Inverse; Semigroups; Primitive Abundant Semigroups; Rees Matrix Semigroups; Homomorphism





Rees matrix Semigroup (A completely O-Simple Semigroup)

Let G a simple group with identity e

P matrix (pm with entries from the zero group G0, where

G0 (GỤ{0} i.e. a L I matrix

Let S = (GxIxL)u {0}



Then S thus constructed is completely O - simple, and S Mo [G,I,L,P] is the I L, Rees matrix semigroup over the O-group Go, with the regular sandwich matrix P.



The PA - blocked Rees 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 0

        "a s/ {0}; aS = Sa = S

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


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



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 , then the set , simple, is called the tensor Product over S, of the two S-systems

* using these definition we make the following 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



Also, if x H* f, then Fx H* f and ex = x


If x1, x2 H* f and fx1 = fs2, then


From (a) and (b) /H*f/ = /H* ef/

Similarly, /H* ef/ = H*el

and 2 is proved.


Proof of Proposition 3



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.



Construction of a Primitive Abundant Semigroup S, with Zero



Main Theorem


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


Proof of Main Theorem





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



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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