Generating Set of the Complete Semigroups of Binary Relations

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1 Applied Mathematics Published Online January 06 in SciRes Generating Set of the Complete Semigroups of Binary Relations Yasha iasamidze Neset Aydin Ali Erdoğan 3 Shota Rustavelli University Batumi Georgia Çanakkale Onsekiz Mart University Çanakkale Turkey 3 Hacettepe University Ankara Turkey Received 6 ecember 05; accepted 5 January 06; published 8 January 06 Copyright 06 by authors and Scientific Research Publishing Inc This work is licensed under the Creative Commons Attribution International License (CC BY) Abstract ifficulties encountered in studying generators of semigroup B X of binary relations defined by a complete X-semilattice of unions arise because of the fact that they are not regular as a rule which makes their investigation problematic In this work for special it has been seen that the B which are defined by semilattice can be generated by the set semigroup X { X ( ) } B = B V X = Keywords Semigroups Binary Relation Generated Set Generators Introduction Theorem Let { m } = be some finite X-semilattice of unions and = { } C P P P Pm 0 be the family of sets of pairwise nonintersecting subsets of the set X If φ is a mapping of the semilattice on the family of sets and ϕ ( ) = P for any i = m and ˆ \ i i i 0 m 0 C which satisfies the condition ϕ ( ) = P0 = T then the following equalities are valid: = P P P P = P T ϕ () T ˆ i How to cite this paper: iasamidze Y Aydin N and Erdoğan A (06) Generating Set of the Complete Semigroups of Binary Relations Applied Mathematics

2 Y iasamidze et al In the sequel these equalities will be called formal It is proved that if the elements of the semilattice are represented in the form then among the parameters P i ( i = 0 m ) there exist such parameters that cannot be empty sets for Such sets P i ( 0< i m ) are called basis sources whereas sets P i ( 0 j m ) which can be empty sets too are called completeness sources It is proved that under the mapping ϕ the number of covering elements of the pre-image of a basis source is always equal to one while under the mapping ϕ the number of covering elements of the pre-image of a completeness source either does not exist or is always greater than one (see [] Chapter ) Some positive results in this direction can be found in []-[6] Let P0 P P Pm be parameters in the formal equalities BX ( ) and m = i= 0 t Pi t X\ Pi t ({ t } t ) () m = Pi t X i= 0 t Pi (( \ ) ) (3) The representation of the binary relation of the form and will be called subquasinormal and maximal subquasinormal If and are the subquasinormal and maximal subquasinormal representations of the binary relation then for the binary relations and the following statements are true: a) BX ( ) ; b) ; c) the subquasinormal representation of the binary relation is quasinormal; d) if P0 P Pm P0 P Pm then is a mapping of the family of sets { } e) if : X \ Results C = P P P Pm in the X-semilattice of unions 0 { } m = t = t for all t X \ then is a mapping satisfying the condition Proposition Let B ( ) X m = Pi t t t i= 0 t Pi t X\ Then = = ({ } ( )) Proof It is easy to see the inclusion holds since x y xy X then xz m y for some z X So z X \ since z x z P t y Then ( ) for some k ( 0 k m ) If and ( i= 0 ( i t P )) i z P t y k ie z P t P k k and y z t For the last condition t P k follows that z y We have xz y and x( ) y Therefore the inclusion is true Of this and by inclusion follows that the equality = = holds Corollary If δ BX ( ) δ then = δ = Proof We have δ and δ Of this follows that = δ = since = Let the X-semilattice = { 3 } of unions given by the diagram of Figure Formal equalities of the given semilattice have a form: 99

3 Y iasamidze et al Figure iagram of = P0 P P P3 P = P0 P P3 P = P0 P P3 P 3 = P0 P P P = P P 0 The parameters P P P 3 are basis sources and the parameters P0 P are completeness sources ie X 3 Example 3 Let X = { 3 567} P = { } P = { 5} P 3 = { 3 6} P0 = P = Then for the for- = 5 = { } = { 3 6} = { 356} mal equalities of the semilattice follows that { } 3 5 = { 3 56} = {{ }{ }{ 3 }{ 3 }{ 3} } and = ({ } { }) ({ } { }) ({ } { }) ({ } { }) Then we have: t = { } t = { } t = { } t P t P t P3 t = t = ; t P0 t P ({ } { }) ({ } { }) ({ } { }) ({ } { }) = ; ({ } { }) ({ } { }) ({ } { }) ({ } { }) = Theorem Let the X-semilattice = { 3 } of unions given by the diagram of Figure B = { BX ( ) V ( X ) = } and X \ 3 Then the set B is generating set of the semigroup BX ( ) Proof It is easy to see that X 6 since P P P3 { } and X \ 3 Now let be any binary rela- B ; n B = n = 3 n Then the equality = = ( is subquasinormal representation of a binary relation ) is true By assumption BX ( ) ie the quasinormal representation of a binary relation have a form Y Y Y Y = Y tion of the semigroup X Of this follows that = (5) ( Y ) ( Y3 3) ( Y ) ( Y ) ( Y0 ) ( Y ) ( Y ) ( Y ) ( Y ) ( Y ) For the binary relation we consider the following case V X = Then = X T where T By element T we consider the following cases: a) Let T In this case suppose that P P P P P T T T 0 3 X on the set { } \{ } T Then 3 () 00

4 where Y Y Y Y { } ( P0 ) ( P T) ( P T) ( P3 T) ( P ) { t } ( t ) = ( Y ) ( Y ) ( Y ) ( Y ) ( Y ) then it is easy to see that B 3 formal equality and equalities (6) and (5) we have: = P P = P P = T = T; since ( ) Y iasamidze et al (6) V = V = From the = P P P P = P P P P = T T = T; = P P P P = P P P P = T T = T; = P0 P P3 P = P0 P P3 P = T T = T; = P P P P P = P P P P P = T T T = T; = since 3 0 ( Y ) ( Y3 3) ( Y ) ( Y ) ( Y0 ) ( Y T) ( Y3 T) ( Y T) ( Y T) ( Y0 T) X T = Y Y Y Y Y = X T = In this case suppose that P0 P P P3 P = 3 are mapping of the set X \ in the set Then = P P P P P t t where Y Y Y Y { } ( 0 ) ( ) ( ) ( 3 3) ( ) ({ } ) ( Y ) ( Y ) ( Y ) ( Y ) ( Y ) since ( ) then it is easy to see that B 3 V = V = From the formal equality and equalities (7) and (5) we have: = P0 P = P0 P = ; 3 = P0 P P P = P0 P P P = ; = P0 P P3 P = P0 P P3 P = ; = P0 P P3 P = P0 P P3 P = ; = P P P P P = P P P P P = ; b) V ( X ) since V ( X ) cases = = Then Let V ( X ) { T} ( Y ) ( Y3 3) ( Y ) ( Y ) ( Y0 ) ( Y ) ( Y3 ) ( Y ) ( Y ) ( Y0 ) = = X V ( X ) { { } { 3} { } { 3 } { } { } } is X-semilattice of unions For the semilattice of unions V ( X ) = where T { } 3 (7) consider the following Then binary relation has representation of the form 0

5 Y iasamidze et al ( Y ) ( Y T T ) = In this case suppose that where Y Y Y Y { } P0 P P P3 P T T X on the set \{ } T Then ( P0 ) ( P T) ( P ) ( P3 T) ( P ) { t } ( t ) = ( Y ) ( Y ) ( Y ) ( Y ) ( Y ) Y = Y and Y3 Y Y Y0 = Y0 then it is easy to see that B since V = V = From the formal equality and equalities (8) and (5) we have: = P P = P P = = 0 0 = P P P P = P P P P = T = T; = P P P P = P P P P = T T = T; = P P P P = P P P P = T = T; = P P P P P = P P P P P = T T = T; ; (8) = Let V ( X ) = { T } form ( Y T T) ( Y0 ) ( Y ) ( Y3 3) ( Y ) ( Y ) ( Y0 ) ( Y ) ( Y3 T) ( Y T) ( Y T) ( Y0 T) = Y Y Y Y Y T = Y Y T where T { } 3 = In this case suppose that Then binary relation has representation of the where Y Y Y Y { } P P P P P T 0 3 X on the set \ { } ( 0 ) ( ) ( ) ( 3 ) ( ) ({ } ) T Then = P P P T P P t f t ( Y ) ( Y ) ( Y ) ( Y ) ( Y ) Y = YT and Y3 Y Y Y0 = Y0 then it is easy to see that B V = V = From the formal equality and equalities (9) and (5) we have: since (9) 0

6 Y iasamidze et al = P0 P = P0 P = T = T; 3 = P0 P P P = P0 P P P = T = ; = P0 P P3 P = P0 P P3 P = = ; = P0 P P3 P = P0 P P3 P = T = ; = P0 P P P3 P = P0 P P P3 P = T = ; = ( Y ) ( Y3 3) ( Y ) ( Y ) ( Y0 ) = ( Y T) ( Y3 ) ( Y ) ( Y ) ( Y0 ) = ( Y T) ( ( Y3 Y Y Y0 ) ) = ( YT T) ( Y0 ) c) V ( X ) = 3 Then V ( X ) { { } { 3 } { } { } { 3 } { 3 } } since V ( X ) is X-semilattice of unions For the semilattice of unions V ( X ) cases Let V ( X ) = { } ( Y ) ( Y ) ( Y 0 ) Then binary relation has representation of the form = In this case suppose that where Y Y Y Y { } P0 P P P3 P X on the set \ { } Then ( P0 ) ( P ) ( P ) ( P3 ) ( P ) ({ t } f ( t )) = ( Y ) ( Y ) ( Y ) ( Y ) ( Y ) consider the following 3 Y = Y Y = Y and Y3 Y Y0 = Y0 then it is easy to see that B since V = V = From the formal equality and equalities (0) and (5) we have: = ( P0 P) = P0 P = = ; 3 = ( P0 P P P) = P0 P P P = = ; = ( P0 P P3 P) = P0 P P3 P = = ; = ( P0 P P3 P) = P0 P P3 P = = ; = ( P0 P P P3 P) = P0 P P P3 P = = ; = = ( Y ) ( Y3 3) ( Y ) ( Y ) ( Y0 ) = ( Y ) ( Y3 ) ( Y ) ( Y ) ( Y0 ) = ( Y ) ( Y ) ( ( Y3 Y Y0 ) ) = ( Y ) ( Y ) ( Y0 ) V X = Then binary relation has representation of the form Let { 3 } (0) 03

7 Y iasamidze et al ( Y ) ( Y 3 3) ( Y 0 ) = In this case suppose that where Y Y Y Y { } P0 P P P3 P 3 X on the set \ { 3 } Then ( P0 ) ( P 3) ( P ) ( P3 ) ( P ) ({ t } f ( t )) = ( Y ) ( Y ) ( Y ) ( Y ) ( Y ) Y = Y Y3 = Y3 and Y Y Y0 = Y0 then it is easy to see that = = From the formal equality and equalities () and (5) we have: B since 3 ( ) ( ) V V = P P = P P = = 0 0 = P P P P = P P P P = = ; 3 3 = P0 P P3 P = P0 P P3 P = = ; 3 = P0 P P3 P = P0 P P3 P = = ; = ( P0 P P P3 P) = P0 P P P3 P = = ; 3 = 3 Let V ( X ) = { } ( Y ) ( Y ) ( Y 0 ) ( Y ) ( Y3 3) ( Y ) ( Y ) ( Y0 ) ( Y ) ( Y3 3) ( Y ) ( Y ) ( Y0 ) ( ) ( 3 3) ( 0 ) = Y Y Y Y Y = ( Y ) ( Y3 3) ( Y0 ) Then binary relation has representation of the form = In this case suppose that P0 P P P3 P X on the set \ { } Then ( P0 ) ( P ) ( P ) ( P3 ) ( P ) ({ t } f ( t )) = ( Y ) ( Y ) ( Y ) ( Y ) ( Y ) ; where Y Y 3 Y Y { } Y B since = Y Y = Y and Y3 Y Y0 = Y0 then it is easy to see that V = V = From the formal equality and equalities () and (5) we have: () () 0

8 Y iasamidze et al = P0 P = P0 P = = ; 3 = P0 P P P = P0 P P P = = ; = P0 P P3 P = P0 P P3 P = = ; = P0 P P3 P = P0 P P3 P = = ; = P0 P P P3 P = P0 P P P3 P = = ; = = ( Y ) ( Y3 3) ( Y ) ( Y ) ( Y0 ) = ( Y ) ( Y3 ) ( Y ) ( Y ) ( Y0 ) = ( Y ) ( Y ) ( ( Y3 Y Y0 ) ) = ( Y ) ( Y ) ( Y0 ) V X = Then binary relation has representation of the form Let { } ( Y ) ( Y ) ( Y 0 ) = In this case suppose that P0 P P P3 P \ Then X on the set { } ( P0 ) ( P ) ( P ) ( P3 ) ( P ) ({ t } f ( t )) = ( Y ) ( Y ) ( Y ) ( Y ) ( Y ) where Y Y 3 Y Y { } Y Y Y B since V V = = Y and Y3 Y Y0 = Y0 then it is easy to see that = = From the formal equality and equalities (3) and (5) we have: = P0 P = P0 P = = ; 3 = P0 P P P = P0 P P P = = ; = P0 P P3 P = P0 P P3 P = = ; = P0 P P3 P = P0 P P3 P = = ; = P0 P P P3 P = P0 P P P3 P = = ; = = ( Y ) ( Y3 3) ( Y ) ( Y ) ( Y0 ) = ( Y ) ( Y3 ) ( Y ) ( Y ) ( Y0 ) = ( Y ) ( Y ) ( ( Y3 Y Y0 ) ) = Y Y Y ( ) ( ) ( 0 ) 5 Let V ( X ) = { 3 } ( Y ) ( Y 3 3) ( Y 0 ) Then binary relation has representation of the form = In this case suppose that P0 P P P3 P 3 3 (3) 05

9 Y iasamidze et al X on the set { } ( 0 ) ( 3) ( ) ( 3 3) ( ) { } \ 3 Then = P P P P P t f t ( Y ) ( Y ) ( Y ) ( Y ) ( Y ) where Y Y 3 Y Y { } Y Y Y B since V V = = Y3 and Y3 Y Y0 = Y0 then it is easy to see that = = From the formal equality and equalities () and (5) we have: = P0 P = P0 P = = ; 3 = P0 P P P = P0 P P P = 3 = ; = P0 P P3 P = P0 P P3 P = 3 3 = 3; = P0 P P3 P = P0 P P3 P = 3 = ; = P0 P P P3 P = P0 P P P3 P = 3 3 = ; = = ( Y ) ( Y3 3) ( Y ) ( Y ) ( Y0 ) = ( Y ) ( Y3 ) ( Y 3) ( Y ) ( Y0 ) = ( Y ) ( Y 3) ( ( Y3 Y Y0 ) ) = ( Y ) ( Y3 3) ( Y0 ) V X = Then binary relation has representation of the form 6 Let { 3 } ( Y ) ( Y 3 3) ( Y 0 ) = In this case suppose that P0 P P P3 P 3 3 \ Then X on the set { 3 } ( P0 ) ( P 3) ( P ) ( P3 3) ( P ) { t } f ( t ) ( Y ) ( Y ) ( Y ) ( Y ) ( Y ) = = where Y Y 3 Y Y { } Y Y Y B since V V = = Y3 and Y3 Y Y0 = Y0 then it is easy to see that = = From the formal equality and equalities (5) and (5) we have: = P0 P = P0 P = = ; = P P P P = P P P P = = ; = P0 P P3 P = P0 P P3 P = 3 3 = 3; = P P P P = P P P P = = ; = P0 P P P3 P = P0 P P P3 P = = ; 3 3 = ( Y ) ( Y3 3) ( Y ) ( Y ) ( Y0 ) ( Y ) ( Y3 ) ( Y 3) ( Y ) ( Y0 ) ( Y ) ( Y 3) (( Y3 Y Y0 ) ) ( Y ) ( Y3 3) ( Y0 ) = = () (5) 06

10 7 Let V ( X ) = { 3 } ( Y ) ( Y 3 3) ( Y ) ( Y 0 ) Then binary relation has representation of the form = In this case suppose that P0 P P P3 P 3 X on the set \ { 3 } Then ( P0 ) ( P 3) ( P ) ( P3 ) ( P ) { t } f ( t ) ( Y ) ( Y ) ( Y ) ( Y ) ( Y ) = where Y Y 3 Y Y { } Y Y Y that B since References Y iasamidze et al = 3 = Y3 Y = Y and Y Y0 = Y0 then it is easy to see V = V = From the formal equality and equalities (6) and (5) we have: = P0 P = P0 P = = ; 3 = P0 P P P = P0 P P P = 3 = 3; = P0 P P3 P = P0 P P3 P = 3 = ; = P0 P P3 P = P0 P P3 P = = ; = P0 P P P3 P = P0 P P P3 P = 3 = ; = = ( Y ) ( Y3 3) ( Y ) ( Y ) ( Y0 ) = ( Y ) ( Y3 3) ( Y ) ( Y ) ( Y0 ) = ( Y ) ( Y3 3) ( Y ) ( ( Y Y0 ) ) = Y Y Y Y ( ) ( 3 3) ( ) ( 0 ) [] iasamidze Ya and Makharadze Sh (03) Complete Semigroups of Binary Relations Cityplace Kriter Country- Region Turkey 50 p [] avedze MKh (968) Generating Sets of Some Subsemigroups of the Semigroup of All Binary Relations in a Finite Set Proc A I Hertzen Leningrad State Polytechn Inst (In Russian) [3] avedze MKh (97) A Semigroup Generated by the Set of All Binary Relations in a Finite Set XIth All-Union Algebraic Colloquium Abstracts of Reports Kishinev 93-9 (In Russian) [] avedze MKh (968) Generating Sets of the Subsemigroup of All Binary Relations in a Finite Set oklady AN BSSR (In Russian) [5] Givradze O (00) Irreducible Generating Sets of Complete Semigroups of Unions BX of Class ( X ) (6) efined by Semilattices Proceedings of the International Conference Modern Algebra and Its Aplications Batumi [6] Givradze O (0) Irreducible Generating Sets of Complete Semigroups of Unions B X efined by Semilattices of Class in Case When X = and \ 3 > Proceedings of the International Conference Modern Algebra and Its Aplications Batumi 07