Idempotent and Regular Elements of the Complete Semigroups of Binary Relations of the Class ( )

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1 Applied Mathematics Published Online February 205 in SciRes. Idempotent and Regular Elements of the Complete Semigroups of Binary Relations Σ 9 of the Class Barış Albayrak Neşet Aydın Çanakkale Onsekiz Mart University Çanakkale Turkey balbayrak77@gmail.com neseta@comu.edu.tr 3 Received 3 January 205; accepted February 205; published 5 February 205 Copyright 205 by authors and Scientific Research Publishing Inc. This work is licensed under the Creative Commons Attribution International License (CC BY). Abstract In this paper we take Q subsemilattice of and we will calculate the number of right unit idempotent and regular elements of B (Q ) satisfied that V ( ) = Q for a finite set. Also we will give a formula for calculate idempotent and regular elements of B (Q) defined by an -semilattice of unions. Keywords Semilattice Semigroup Binary Relation. Introduction Let be a nonempty set and B be semigroup of all binary relations on the set. If is a nonempty set of subsets of which is closed under the union then is called a complete -semilattice of unions. Let f be an arbitrary mapping from into. Then one can construct a binary relation f on by = x f x B and called a complete semi- f ({ } ). The set of all such binary relations is denoted by x group of binary relations defined by an -semilattice of unions. We use the notations y = { x yx} Y = y V ( ) { Y Y } A representation of a binary relation of the form = ( YT T) y Y T V ( ) = YT { y y T} = =. is called quasinormal. Note that How to cite this paper: Albayrak B. and Aydın N. (205) Idempotent and Regular Elements of the Complete Semigroups of Binary Relations of the Class Σ 3 ( 9). Applied Mathematics

2 if = ( YT T) B. Albayrak N. Aydın is a quasinormal representation of the binary relation then YT YT = for T T V( ) T V( ) and T T. A complete -semilattice of unions is an I-semilattice of unions if Λ( t ) for any t and Z= Λ ( t ) for any nonempty element Z of. t Z is said to be right unit if β = β B B is idempotent if is said to be regular if β = for some β B ( ). Let ' '' be complete -semilattices of unions and ϕ be a one-to-one mapping from ' to ''. A mapping ϕ : ϕ = ϕ T for all nonempty subset of the se- Now B ( ) =. And B ( ) is a complete isomorphism provided for all β. Also milattice '. Besides that if ϕ: V( ) is a complete isomorphism where B ( ) ( T) for all T V( ) ϕ is said to be a complete -isomorphism. T ϕ = T Let Q and ' be respectively some I and -subsemilattices of the complete -semilattice of unions. Then where ϕ : Q where { } ( ) = Rϕ Q B regular element ϕ complete -isomorphism complete isomorphism and V ( ) = Q. Besides let us denote RQ ( ) = R ( Q ) and R( ) R( Q ) ϕ Φ( Q ) ϕ = Q Ω( Q) ( Q ) { ϕ ϕ: Q is a complete -isomorhism B ( ) } Φ = ( Q) { Q Q is I-subsemilattices of which is complete isomorphic to Q} Ω = This structure was comprehensively investigated in iasamidze []. Lemma. [] If Q is complete -semilattice of unions and I( Q ) is the set all right units of the semigroup B ( Q ) then I( Q) = Rid Q ( QQ ). Lemma 2. [2] Let be a finite set be a complete -semilattice of unions and Q = T T T T T T T T be -subsemilattice of unions of satisfies the following conditions { } Q is I-semilattice of unions T \ T T \ T T \ T T \ T T T = T T T = T T = is -iso- Theorem. [2] Let be a finite set and Q be I-semilattice. If { T T2 T3 T4 T5 T6 T7 T8} morphic to Q and Ω ( Q) = m0 then T2\ T T4\ T3 T4\ T3 T3\ T4 T3\ T4 m0 4 ( 2 ( 2 ) ) ( 3 2 ) ( 3 2 ) 5 T7\ T6 T7\ T6 T6\ T7 T6\ T7 \ T8 ( 6 5 ) ( 6 5 ) 8. T3 T4 \ T T7 T6 \ T5 R = Theorem 2. [2] Let B ( Q) ( ) = Q. B ( ) 8 be a quasinormal representation of the form = ( Yi Ti) such that V is a regular iff for some complete -isomorphism ϕ : Q the following conditions are satisfied: Y ϕ( T) Y Y2 ϕ( T2) Y Y2 Y3 ϕ( T3) Y Y2 Y4 ϕ( T4) Y Y2 Y3 Y4 Y5 Y6 ϕ( T6) Y Y2 Y3 Y4 Y5 Y7 ϕ( T7) Y2 ϕ( T2) Y ϕ T Y ϕ T Y ϕ T Y ϕ T

3 B. Albayrak N. Aydın Let be a finite set and { T T T T T T T T T } = be a complete -semilattice of unions which satisfies the following conditions T \ T T \ T T \ T T \ T T \ T T \ T T T 2 = T 3 T 4 T 5 = T 6 T T = T T T we denote the class of all complete - 3 semilattice of unions whose every element is isomophic to an -semilattice of the form. All subsemilattice of = { T T2 T3 T4 T5 T6 T7 T8 are given in Figure 2. In iasamidze [] it has shown that subsemilattices - 5 are I-semilattice of unions and subsemilattices 7-24 are not I-semilattice of unions. In Yeşil Sungur [3] and Albayrak [4] they have shown that subsemilattices 25 and 26 are I-semilattice of unions if and only if T T2 =. Also they found that number of right unit idempotent and regular elements in subsemilattices. In this paper we take in particular Q = { TT 3 T4 T5 T6 T7 T8 subsemilattice of. We will calculate the number of right unit idempotent and regular elements of B ( Q ) satisfied that V( ) = Q for a finite set. Also we will give a formula for calculate idempotent and regular elements of B ( ) defined by an = T T T T T T T T T. The diagram of the is shown in Figure. By the symbol ( ) -semilattice of unions { } Figure. iagram of. 34

4 B. Albayrak N. Aydın 2. Results Let Q { T T T T T T T T } = be complete -subsemilattice of satisfies the following conditions T \ T T \ T T \ T T \ T T T = T T T = T T. The diagram of the Q is shown in Figure 3. From Lemma 2 Q is I-semilattice of unions. Let Qϑ I denote the set of all I-subsemilattice of the semilattice which are isomorphic of the -semilattice Q. Then we get Let B ( Q ) {{ }{ }} Q ϑ = T T T T T T T T T T T T T T T T I be a idempotent element having a quasinormal representation of the form Figure 2. All subsemilattice of. Figure 3. The diagram of the Q. 35

5 B. Albayrak N. Aydın 9 ( YT T) ( Yi Ti) = B Q. such that V ( ) = Q 3 ( ) Lemma 3. If is a finite set and ( ) B Q then the number I Q may be calculated by formula: I Q. First we calculate number of this idempotent elements in I Q is the set all right units of the semigroup T3\ T ( T5 T4) \ T3 T5\ T4 T5\ T4 T4\ T5 T4\ T5 ((2 ) 2 ) ( 3 2 ) ( 3 2 ) = T7 T8 \ T6 T8\ T7 T8\ T7 T7\ T8 T7\ T8 \ T Proof. From Lemma we have I( Q ) R ( Q Q ) = where id Q For this reason = Q in Theorem. Then we obtain T3\ T T5 T4 \ T3 T5\ T4 T5\ T4 T4\ T5 T4\ T5 ( 2 2 ) ( 3 2 ) ( 3 2 ) I Q Theorem 3. If is a finite set and I ( Q ) then the number = T7 T8 \ T6 T8\ T7 T8\ T7 T7\ T8 T7\ T8 \ T id Q is identity mapping of the set Q. is the set all idempotent elements of the semigroup I Q may be calculated by formula: T5 T4 \ T3 I Q = T3\ T2 T5\ T4 T5\ T4 T4\ T5 T4\ T5 ( T7 T8) \ T6 T8\ T7 T8\ T7 T7\ T8 T7\ T8 \ T9 T3\ T ( T5 T4) \ T3 T5\ T4 T5\ T4 T4\ T5 T4\ T5 (( 2 ) 2 ) ( 3 2 ) ( 3 2 ) ( T7 T8) \ T6 T8\ T7 T8\ T7 T7\ T8 T7\ T8 \ T9 ( 6 5 ) ( 6 5 ) Proof. By using Lemma 3 we have number of right units of the semigroup B ( Q ) defined by Q = { T T3 T4 T5 T6 T7 T8 for T { T T2}. Then number of idempotent elements of I ( Q ) by formula I ( Q ) = I( ). By using QϑI we obtain above formula. {{ }{ }} Q ϑ = T T T T T T T T T T T T T T T T I Now we will calculate number of regular elements B ( Q ) 9 form = ( YT T) ( Yi Ti) B Q calculated having a quasinormal representation of the such that V ( ) = Q. Let R ( Q ) 3 semigroup B ( Q ). By using Q I {{ T T3 T4 T5 T6 T7 T8 { T2 T3 T4 T5 T6 T7 T8 } Ω ( Q ) = 2. The number of all automorphisms of the semilattice Q is q = 4. These are be the set all regular elements of the ϑ = we get TTTTTTTT TTTTTTTT IQ = ϕ = TTTTTTTT TTTTTTTT TTTTTTTT TTTTTTTT θ = τ = TTTTTTTT TTTTTTTT Then Φ ( Q ) = 4. Also by using = { T2 T3 T4 T5 T6 T7 T8 2 = { T2 T3 T5 T4 T6 T7 T8 3 = { T2 T3 T4 T5 T6 T8 T7 4 = { T2 T3 T5 T4 T6 T8 T7 5 = { T T3 T4 T5 T6 T7 T8 6 = { T T3 T5 T4 T6 T7 T8 = { T T T T T T T T } = { T T T T T T T T }

6 B. Albayrak N. Aydın we get R ( Q ) R( ) 8 =. Theorem 4. If is a finite set and R ( Q ) the number R Q R i may be calculated by formula: B Q then is the set all regular elements of the semigroup T3\ T2 T5\ T4 T5\ T4 ( Q ) = 42 ( 2 ) ( ) ( ) ( ) T3\ T T5\ T4 T5\ T ( 2 ) ( ) ( ) ( ) T5 T4 \ T3 T7 T8 \ T6 T4\ T5 T4\ T5 T8\ T7 T8\ T7 T7\ T8 T7\ T8 \ T T5 T4 \ T3 T7 T8 \ T6 T4\ T5 T4\ T5 T8\ T7 T8\ T7 T7\ T8 T7\ T8 \ T Proof. To account for the elements that are in R ( Q ) we first subtract out intersection of ( i ) R( ) R( 2). By using Theorem 2 and Q = { T T3 T4 T5 T6 T7 T8 R( ) R( ) R( ) and R( ) 2 2 Y T Y Y T Y Y Y T T 2 T 3 3 T Y Y Y T Y Y Y Y Y Y T T T Y Y Y Y Y Y T Y T T Y T Y T Y T Y T Y T Y Y T Y Y Y T T 2 T 3 3 T Y Y Y T Y Y Y Y Y Y T T T Y Y Y Y Y Y T Y T T Y T Y 4 T 5 Y 7 T 7 Y 8 T We get Y4 T4 Y4 ( YT Y3 Y5 ) R R 2 =. Smilarly R( i) R( j) R s. Let which is a contradiction with Y 4 Y T Y 3 Y 5 are dis- = for i j = 6. Thus we obtain joint sets. Then R ( Q ) = R( ) + R( ) + R( ) + R( ) + R( ) + R( ) + R( ) + R( ) From Theorem we get above formula. Corollary. If is a finite set I is the set all idempotent elements of the semigroup B set all regular elements of the semigroup B ( ) then the number I and and R is the R may be calculated by formula: ( I) I = ( Qi) ( R) = R ( Qi) Proof. Let I be the set of all idempotent elements of the semigroup B element of B of. I ( Q i ) is given in iasamidze [] for ( 2 5) elements of the subsemigroup ( I ) I ( Q ). Then number of idempotent is equal to sum of idempotent elements of the subsemigroup defined by I-subsemilattice i =. From Theorem 3 we have number of idempotent B Q. Then the number I may be calculated by formula = i. Similarly the number References R may be calculated by formula ( R ) R ( Q ) =. [] iasamidze Ya. and Makharadze Sh. (203) Complete Semigroups of Binary Relations. Kriter Yayınevi İstanbul 524 p. [2] Albayrak B. Aydin N. and iasamidze Ya. (203) Reguler Elements of the Complete Semigroups of Binary Relations of the Class ( 8). International Journal of Pure and Applied Mathematics i 37

7 B. Albayrak N. Aydın [3] Yeşil Sungur. and Aydin N. (204) Reguler Elements of the Complete Semigroups of Binary Relations of the Class ( 7). General Mathematics Notes [4] Albayrak B. Aydin N. and Yeşil Sungur. (204) Regular Elements of Semigroups B efined by the Generalized -Semilattice. General Mathematics Notes

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