Idempotent  semigroups Текст научной статьи по специальности «Математика»

Научни трудове на Съюза на учените в България-Пловдив, серия Б. Естествени и хуманитарни науки, т. XV, 2013 г. Научна сесия „Техника и технологии, естествени и хуманитарни науки", 25-26 X 2012 Scientific researches of the Union of Scientists in Bulgaria-Plovdiv, series B. Natural Sciences and the Humanities, Vol. XV, ISSN 1311-9192, Technics, Technologies, Natural Sciences and Humanities Session, 25-26 oktober 2012.

IDEMPOTENT Г-SEMIGROUPS

Sabri Sadikua a-FGJT; University of Prishtina, Mitrovica, Republic of

Kosovo

sabsadiku@hotmail.com

Abstract: In this paper, we have studied some characteristics of idempotent r - semigroup.

First we have defined the r - semigroup and idempotent r - semigroup. In addition, we defined r - subsemigroup and idempotent r - subsemigroup.

Therefore, we take Theorem 1, which provides that every r - subsemigroup of idempotent r - semigroup S is also idempotent r - semigroup. This theorem will be illustrated by Example 4.

In this paper, set of all a - idempotent in r - semigroup is noted by Ea.

Furthermore, we defined super idempotent or general idempotent in idempotent r-semigroup. In addition, Theorem 3 shows that union of all sets of idempotent is equal by S, exactly

S =UEa .

aer

Key words: r - semigroup; idempotent r - semigroup; idempotent r - subsemigroup,

Introduction: Theory of r - semigroup is expanded in a natural way while its foundation is found in the theory of semigroups. Therefore, the theory of idempotent r - semigroups finds its foundation and has expanded by the theory of idempotent semigroups. Below, we define the r -algebraic structure and r - groupoid.

Definition 1. Let

S = (a, b, c,...}

and

r = (a, fi,y,...}

two nonempty sets. A mapping f : S xfxS ^ S or f : (a ,a,b) ^ c ; a, b, c e S ;

a er called ternary operation in S and r . This operation we denote by ()r or by (+) r . The element (a, a, b) we simply denote by aab . Operation f is commutative if Va, b e S , a er satisfies condition : aab = baa.

Operation f is associative if it satisfies condition : (aab )Pc = aa(bfic) ; Va, b, c eS ,a, ¡3 er .

According [2] we now have defined r - algebraic structure and r - groupoid.

Definition 2. Let

S = {a, b, c,...} and

r = {a, ¡,g,...}

two nonempty sets . Order pair (S,()r ) is called r - algebraic structure.

Definition 3. A r — algebraic structure (S,()r) is called r— groupoid if it satisfies condition:

(i) Va, b eS ,a, ¡3 er ^ aab e S . Example 1. Let

S = {a = 4z + 3, z e Z}= {••• -13, -9, -5, -1,3,7,11,15,- •} and

r = {a = 4z +1,z e Z} = {• -11,-7,-3,1,5,9,13,-•}

two sets. If a = 4 z1 + 3, b = 4 z2 + 3 and a = 4 z3 +1 where a, b e S and a er . Then aab = 4z1 + 3 + 4z3 +1 + 4z2 + 3 = 4(z1 + z2 + z3 +1) + 3 = 4z + 3 e S

Therefore (S,(+)r) is r- groupoid , where operation (+)r or aab is addition of integers .

Now we defined r- semigroup S according [1]

Definition 4. A r — algebraic structure (S,()r) is called r— semigroup if it satisfies condition:

(i) Va, b eS, aer ^ aab e S .

(ii) V a, b, c e S ; a, /er ( aab )/c = aa(b/c ). r — semigroup we can define it also in this way:

A r — groupoid (S,(• )r ) satisfying the associative law

( aab)/c = aa( b/c) ; ( Va, b, c e S ; a, / eT) is a r — semigroup.

Example 2. Let S be the set of all matrices of type:

[Y a 0 ^

S = ■

v b 1J

: a, b e R '

and r be the set of all matrices:

r =

rX 0^

v0 1J

: x e R }.

We can prove that the system ( S,(• )r ) is r - semigroup, where operation ()r is the product of the matrices.

Definition 5. An element e e S is said to be an idempotent of r - semigroup S if eae = e for some a er . In this case we call e an a - idempotent. S is a idempotent r -semigroup if and only if every element of S is idempotent .

Set of all elements a - idempotent note by Ea, a er .

Example 3. Let

S = E =

and

f10 ^ f 0 0 ^ f 0 0 ^

v0 0y

E =

' 2

v0 1 y

E =

v0

r = < a =

f 1 0^

v0 1 y

two sets. Then (S ,(-)r ) is idempotent r - semigroup, where operation ()r is the product of the matrices.

Lemmai. If Sis a idempotent semigroup, r = (a = 1} and alb = ab, then S is idempotent r - semigroup.

Proof. Let a e S , then aaa = a1 a = aa = a2 = a, Va e S, a = 1 e r. Then S is idempotent r - semigroup.

From Lema 1. we can conclude that every idempotent semgroup S may be considered as idempotent

r - semigroup S .

Definition 5. Let S be a r - semigroup. Subset M of S is r - subsemigroup of r - semigroup S if MFM c M , where MrM = {man: m, n e M; a e r}.

Theorem 1. Every r - subsemigroup of idempotent r - semigroup S is idempotent r -semigroup.

Proff. If S1 is r - subsemigroup of r - idempotent semigroup S, then for every a e S1 ^ a e S , but S is idempotent r - semigroup then 3 a er for which is aaa = a .

This proves that S1 is idempotent r - subsemigroup. To illustrate this theorem, let us consider the following example.

Example. 4. Let S be the set

S = , A2, A3, A4, A5} where

A, =

and

(10 ^ ( 0 1 ^ ( 0 0 ^ ( 0 0 ^ ( 0 0 ^

4 —

V0 0y

> A2 =

V0 0y

> A3 =

V1 0y

, A4 =

v0 1 y

V0 0y

r —

(1 0^ (0 1^

V0 1 y

V1 °y

— {a, P}.

and

Then S is a idempotent r - semigroup, where set of a - idempotent is Ea = {Aj, A4, A5}

set of P - idempotent is EP = {A, A3, A5} .

If S1 = {Aj, A2, A5} c S then

A}aA} — Aj, A_2a Aj — A5, — A5,

A^aA.2 — A2, A2aA2 — A5, A5aA2 — A5,

A.jaA.5 - A5 ; A2^A45 - A5 ; ^5^^4.5 - A5 ;

AjPAj = A5; A2PAx — Aj; A,PAX - A;

-AjPA-2 - A5 ; .A2P-^2 - A2 ; A5PA2 - A5 ;

AXPA5 - A5; A2PA5 — A ; A5PA5 — A .

We conclude that Sj is idempotent r - subsemigroup of idempotent r - semigroup S.

Element A5 e Ea and A5 e A5 e Ea o Ep . Element A5 e S we call super idempotent.

Definition 6. Element a e S we call super idempotent in r - semigroup S if a e Ea .

aer

Following theorem determines more accurately the super idempotent element a in r -semigroup S.

Theorem 2. If S is a idempotent r - semigroup and a element of S such that a e Ea then aer

element a e S is super idempotent.

Proof. Straightforward.

In order to find connection between r - idempotent semigroup S and all sets of idempotent

E

a the following theorem is formulated.

Theorem 3. If S is a r — idempotent semigroup, then

S = U Ea

E„

a

Prof. If a e S , then aaa = a , for a er . Then a e Ea. Every element of S belongs for a er from this implies

S Ç U Ea..........d)

aer

Conversely.

If X e Ex, then eF, xPx = x and x e S . From this we conclude that

S 3 U Ea .......(2)

aer

From (1) and (2) we conclude that S = U Ea .

aer

Conclusion: We conclude that every subsemigroup of idempotent r— semigroup is idempotent r — subsemigroup and if super idempotent element a exists in

idempotent r — semigroup then a e Ea .

aer

References

[1] Sen M. K.,Saha N. K., On r— semigroups I, Bull.Cal.Math.Soc.78 (1986), 180-186

[2]S.Sadiku, Necessary and Sufficient Conditions Where One r — semigroups is a r — group, Journal of Modern Mathematics and Statistics, 4(1): 44-49, 2010, ISSN:1994-5388

[3]SahaN.K., On r— semigroups II, Bull.Cal.Math.Soc.79 (1987), 331-335

[4]M. Petrich, Inverse Semigroups, ISBN 0-471-875445-7, 1984 BY John Wiley&Sons.