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1 International Journal of Engineering Science and Innovative echnology (IJESI) Volume Issue 3 May 201 Finite semigroups of binary relations defined by semi lattices of the class Σ( ) Mzevinar Bakuridze 1 and Alexander Bakuridze 2 Faculty of Physics Mathematics and Computer Sciences epartment of Mathematics Shota Rustaveli Batumi State U n i v e r s i t y 3 Ninoshvili St Batumi 010 Georgia ABSRAC: he paper deals with the structure of idempotent and maximal subgroups of finite semi groups of binary relations that are identified by the semi lattices of class counting the number of idempotent AMS Subject Classification: 20M0 Key words: semi lattice semi group of binary relations For finite set we found the formula of I INROUCION Let be any non-empty set is a non-empty set of subsets of the set closed in relation to the operation set-theoretic union of elements from f is a map of set in the set For each such map f we will compare binary relation f x f x x Set of all such kind f f on the set satisfying will be denoted by B It is easy to prove that B f : is subsemigroup of semi group B which is called finite semi groups of binary relations defined with semi lattice of union Furthermore let Y - be a certain set of subset and be e certain binary relation on the set Let us denote with Y set of all such elements x for which there is an element y Y which x y Moreover V Y Y if и V Y Y Z Z efinition 1 Let be a certain set semi lattice of set andy y y If if if V V V V V if V and B then it is obvious that any binary relation of subgroup Y (1) V can always be represented as B 29

2 International Journal of Engineering Science and Innovative echnology (IJESI) Volume Issue 3 May 201 In the future such a representation of a binary relation we shall call quasinormal and call normal if Y for any V efinition 2 Let is a finite semilattice of union and N Z Z Z for any Z N is the set of all lower bounds It is obvious that of non-empty subsets from in the semilattice If N then N exact lower bound of the set in Let us denote this element with N Note if the element in the semi lattice exists then we can write is the ie otherwise - efinition 3 If t и Z t Z It is said that finite semi lattice of union is t I semi lattice of union if it satisfies the following two conditions: a) for anyt t t for any non-empty element Z of semi lattice tz b) Z heorem 1 Binary relation B idempotent and V is a right unit of given semigroup unit if and only if is heorem 2 If is finite semilattice of union Semigroup B I semilattice of union has a right unit if and only if is heorem 3 Let E r and I respectively denote the finite semilattice of union set of all I subsemigroup of semilattice the set of all the right units of the semigroup B and the set of all idempotent of the semigroup B hen sets E r and I the following statements are true: a) if и then r r E E for any elements and sets 1) ; r 2) I E ; r 3) for a finite set Х the following equality is true I E b) If then 1) E ; r 2) I E ; r r E for any elements and of set satisfying the condition satisfying the condition 30

3 International Journal of Engineering Science and Innovative echnology (IJESI) Volume Issue 3 May 201 r 3) for the finite Х the following equality is true I E heorem 4 Let 1 Q 0 1 m m there is a semilattice which 0 1 m relation of semigroup hen binary m B Q having a quasi-normal representation of the form Yi i Q V Q is a right unit of semigroup for any p01 m 1 and q 12 m heorem Let number that 0 j m 3 and m p i0 p that B Q if and only if Y0 Y1 Y и Y Q m be such a semilattice and j such a fixed natural Binary relation of semigroup B m Yi i that Q V Q i0 0 1 j j 1 j3 m 0 1 j j2 j3 m \ \ j1 j2 j2 j1 j1 j2 j3 Q having a quasi-normal representation of the form is a right unit of semigroup B Q if and only if Y0 Y1 Yj j 1 j2 0 1 j j2 j2 Y0 Y1 Yp p и Y q For any p012 m 1 q 12 m Let and Y Y Y Y q p j 2 q j 3 respectively are certain non-empty set and such a class -semilattice of union each element of which is isomorphic of some -semilattice of union Z Z Z Z Z } satisfying the following conditions: Z Z1 Z Z4 Z2 Z Z3 Z1 \ Z2 Z2 \ Z1 Z1 \ Z4 Z4 \ Z1 Z \ Z Z \ Z Z \ Z Z \ Z Semi lattices satisfying conditions (1) is given on the figure Fig 1 Further we will assume that ( ) { P P4 P3 P2 P1 P0 } semilattices on the set C () that ( ) P0 and Z3 { C is a set of and there is such a map of q q Z 1 Z Z 2 4 Z Fig 1 31

4 International Journal of Engineering Science and Innovative echnology (IJESI) Volume Issue 3 May 201 ( Z ) for any i 1234 i P i following form: hen the formal equality for a given semi lattices elements have the P0 P1 P2 P3 P4 P Z1 P0 P1 P2 P3 P4 P Z2 P0 P1 P3 P4 P Z3 P0 P1 P2 P4 P Z4 P0 P1 P3 P4 P Z P 0 where P 1 1 P 1 2 P 3 1 P 4 1 P 0 0 P 0 and 4 ie elements 1 P 4 are basic sources and P 0 and P semi lattice of union (see [2 or 3]) Lemma 1 Let is a finite set n 4 1 (7 n 4 n n 4 4 n 3 n s ) 2 and P P 2 P 3 and s hen Proof By definition of semi lattices of this class we have that number of basic sources 4 Moreover any automorphisms of semilattices of class are excluded bye the mapping of the form Z Z4 Z3 Z2 Z1 Z Z4 Z3 Z2 Z1 1 2 Z Z4 Z3 Z2 Z1 Z Z4 Z1 Z2 Z3 herefore their number is q 2 now considering the theorem 24 from [2] we will have: P1 1 ( 1) C ( P!) i s 2 ( i 1)! ( P 1 i)! P4 i1 Pi1 P3 n m3 where C k j j! After simplifying the last expression we obtain:!! k j k 1 (7 n 4 n n 4 4 n 3 n s ) 2 he lemma is proved Example 1 Let n hen accordingly we will have: s

5 International Journal of Engineering Science and Innovative echnology (IJESI) Volume Issue 3 May 201 B hese numbers show if for example 10 then the number of elements in the class of semigroups in which each element is determined by a semilattice of class of elements in each semigroup belonging to this class equals to 0417 Our goal is to study finite semi groups of binary relations B of the class equals to and the number determined by -semilattice of union It is clear that quazinormal representations of any element of from the semigroup B following form: Y Z Y4 Z4 Y3 Z3 Y2 Z2 Y1 Z1 Y0 (*) has the Y Y Y Y Y and Y 0 are some disjoint subsets of the set where At first we will describe idempotent elements of the subgroup B In order to solve the problem by heorem 113 it is necessary first to identify I semilattice of union (see definition 113) in the semilattices Lemma 2 Let Z Z Z Z Z а) Z Z Z Z Z ; hen the subsets of the form b) Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z c d e Z Z4 Z Z3 Z Z2 Z Z1 Z4 Z3 Z4 Z2 Z4 Z1 Z3 Z2 Z3 Z1 Z2 Z1 Z Z4 Z2 ; Z3 Z2 Z1 Z4 Z3 Z1 Z4 Z2 Z1 Z4 Z3 Z1 Z Z2 Z1 Z Z3 Z1 Z Z3 Z2 Z Z4 Z1 Z Z4 Z3 Z Z4 Z2 ; Z4 Z3 Z2 Z1 Z Z3 Z2 Z1 Z Z4 Z2 Z1 Z Z4 Z3 Z1 Z Z4 Z3 Z2 ; ; 33

6 f ISSN: International Journal of Engineering Science and Innovative echnology (IJESI) Volume Issue 3 May 201 Z Z4 Z3 Z2 Z1 subsemilattice of the semilattice is excluded Proof Obviously all the single-elements of subset of semilattice are its subsemilattices he number of two-element of subsets of semilattices equals C 2 1 hey have the form: { Z Z 4}{ Z Z3}{ Z Z2}{ Z Z1}{ Z }{ Z4 Z3}{ Z4 Z2}{ Z4 Z1} { Z }{ Z Z }{ Z Z }{ Z }{ Z Z }{ Z }{ Z } From these subsets of subsemilattices semilattice of union will be only those sets which are of - chain he number of all three-element sets of -set of unions equals C 3 20 { Z Z4 Z 2}{ Z Z4 }{ Z Z3 }{ Z Z2 }{ Z Z1 } { Z4 Z3 }{ Z4 Z2 }{ Z4 Z1 }{ Z3 Z2 }{ Z3 Z1 }{ Z2 Z1 }; { Z Z4 Z3}{ Z Z4 Z1}{ Z Z3 Z2}{ Z Z3 Z1} { Z Z Z }{ Z Z Z }{ Z Z Z }{ Z Z Z }{ Z Z Z }; It is easy to prove that from the given subsets the 9 sets are not subsemilattices of the semilattice he number of all four elements subsets of semilattice equals C 4 1 hey have the form { Z3 Z2 Z1 }{ Z4 Z2 Z1 }{ Z4 Z3 Z1 }{ Z4 Z3 Z2 }{ Z Z2 Z1 }; { Z Z3 Z1 }{ Z Z3 Z2 }{ Z Z4 Z1 }{ Z Z4 Z2 }{ Z Z4 Z3 } { Z Z Z Z }{ Z Z Z Z }{ Z Z Z Z }{ Z Z Z Z }{ Z Z Z Z } It can be verified that from the given subsets the last 10 sets are not semilattices From the proved it immediately follows that the diagrams of type 2 all diagrams of own sublattices of semilattice are excluded Lemma 3 Let Z Z Z Z Z а) Z Z Z Z Z Fig2 hen the subsets of form: 34

7 International Journal of Engineering Science and Innovative echnology (IJESI) Volume Issue 3 May 201 Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z b) c) Z Z4 Z2 Z Z4 Z Z3 Z Z2 Z Z1 Z4 Z2 d) 4 2 Z Z Z e) Z Z2 Z1 Z Z3 Z1 Z Z3 Z2 Z Z4 Z1 Z Z4 Z3 all I subsemilattices of semilattice are excluded Proof he truth of given lemma follows directly from the theorem 2 from [1] and from the theorem 2 from [4] Lemma is proved From the proved lemma it directly follows that in Fig 3 he given diagrams exclude all diagrams of I -subsemilattice of semi lattice heorem 1 Let Z Z Z Z Z semigroup B conditions listed below: and Z hen binary relations of if and only if is idempotent element of this semi group when it meets at least one of the а) where ; в) ( ) ( ) for some conditions: и ; and satisfying the с) ( ) ( ) ( ) for some and satisfying the conditions: and ; d) ( Z ) ( Z ) ( Z ) ( ) for some satisfying the conditions: Z 4 Z 4 Z 0 и Fig Z2 Z 4 4 3

8 International Journal of Engineering Science and Innovative echnology (IJESI) Volume Issue 3 May 201 ( Z ) ( ) ( ) ( ) for some Z Z Z Z e) 0 и \ \ satisfying the conditions: Z и 0 Proof Let Z Z Z Z Z semi group B 1 he theorem is proved and binary relation is an idempotent element of the hen the truth of the theorem follows directly from Lemma 23 from theorem 14 and heorem 2 Let finite set Z Z Z Z Z denote the set of all idempotent elements of the semi group B I can be calculated by the formula: and Z If by symbol I we then the number of elements I in the set \ Z \ Z4 \ Z3 \ Z2 \ Z1 \ Z4\ Z \ Z4 I Z3 \ Z \ Z3 Z2 \ Z Z1 \ Z \ Z 2 \ Z 2 \ 4 2 \ Z \ Z \ Z \ Z4 \ Z \ 4 4 Z \ Z Z \ Z \ Z Z Z Z3 \ Z 3 3 Z2 \ Z 2 2 Z1\ Z 1 1 Z2\ Z \ 2 \ \ 4 2 \ Z 4 2 \ Z2 \ 2 \ \ Z \ Z \ \ Z \ Z \ \ Z \ Z \ \ Z \ Z \ Z Z Z Z Z Z Z Z Z1\ Z2 Z3 \ Z1 Z1 \ Z3 Z3 \ Z2 Z2 \ Z3 Z4 \ Z1 Z1 \ Z4 Z4 \ Z3 Z3 \ Z4 \ \ \ \ \ 1 4 Proof he truth of the theorem follows directly from heorem from [1] and from heorem 32 from [] REFERENCES [1] Ya I iasamidze Complete Semi groups of Binary Relations Journal of Mathematical Sciences Plenum Publ Cor New York Vol 117 No [2] Ya I iasamidze Sh I Makharadze G J Fartenadze O Givradze On finite semilattices of unions Journal of Mathematical Sciences Plenum Publ Cor New York [3] Я И Диасамидзе Ш И Махарадзе Г Ж Партенадзе О Т Гиврадзе О конечных полурешнтках объединений Современная математика и ее приложения Алгебра и геометрия Тбилиси 200 т [4] iasamidze Ya Makharadze Sh N V Rokva On I Semilattices of Unions Bull Georg Acad Sci [] Makharadze Sh I iasamidze I Ya One class of complete semi groups of binary relations J of Math Sciences Plenum Publ Cor New York