Idempotent Elements of the Semigroups B ( D ) defined by Semilattices of the Class Σ ( X ), when Z 7

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1 IJISE - International Journal of Innovative Science Engineering & echnology Vol Issue January 0 ISSN Idempotent Elements of the Semigroups B ( defined by Semilattices of the Class Σ ( when 7 8 Nino sinaridze ninocinaridze@mailru epartment of Mathematics Faculty of Physics Mathematics and Computer Sciences Shota Rustaveli Batumi State University 5 Ninoshvili St Batumi 00 Georgia ABSRAC he paper gives description of idempotent elements of the semigroup B ( which are defined by semilattices of the class ( for which intersection the minimal elements is not empty When is a finite 8 set the formulas are derived by means of which the number of idempotent elements of the semigroup is calculated 00 mathematical Subject Classification 0M05 Key words: semilattice semigroup binary relation idempotent element Introduction Let be an arbitrary nonempty set be a semilattice of unions ie a nonempty set of subsets of the set that is closed with respect to the set-theoretic operations of unification of elements from f be an arbitrary mapping from into o each such a mapping f there corresponds a binary relation satisfies the condition f = ({ x f ( x to prove that B ( he set of all such f x on the set that f ( f : is denoted by B ( It is easy is a semigroup with respect to the operation of multiplication of binary relations which is called a complete semigroup of binary relations defined by a semilattice of unions (see ([] Item By we denote an empty binary relation or empty subset of the set he condition ( xy written in the form x y Let xy Y will be and t = Y hen by symbols we denote the following sets: By symbol ( t B ( { ( { y = x yx Y = y V = Y Y y Y { { t { { = = t Y = { x x = = = we mean an exact lower bound of the set in the semilattice ε If ε ε = ε efinition Let B ( and ε is called right unit if ε = Y then ε is called an idempotent element of the semigroup B ( for any B ( (see [] [] [] efinition We say that a complete semilattice of unions is an I semilattice of unions if it satisfies the following two conditions: a ( t b = ( for any t [] or [] t for any nonempty element of (see ([] definition ([] definition t

2 IJISE - International Journal of Innovative Science Engineering & echnology Vol Issue January 0 efinition Let B ( V ( ( ( of the form = Y Note that if V following conditions are true: = Y V ISSN and Y = { y y = A representation of a binary relation ( ( Y = Y V ( is colled quasinormal Y = for V( and Let ( m n is a quasinormal representation of a binary relation then the denote the class of all complete semilattice of unions where every element is isomorphic to a fixed semilattice (see [] heorem Let be a complete semilattice of unions he semigroup B ( possesses right unit iff is an I semilattice of unions (see ([] heorem ([] heorem or [5] heorem Let be a finite set and ( be the set of all those elements of the semilattice ( \{ Q = V which are nonlimiting elements of the set Q A binary relation having a quasinormal representation = ( Y is an idempotent element of this semigroup iff V( a ( b Y for any ( c V is complete I semilattise of unions ( Y for any nonlimiting element of the set ( ( r heorem Let Σ ( E ( (see ([] heorem 9 ([] heorem 9 or [5] and I denote respectively the complete semilattice of unions the set of all I subsemilatices of the semilattice the set of all right units of the semigroup B ( all idempotents of the semigroup B ( b if then hen for the sets E ( r ( ( r ( r ( ( = for any elements and of the set ( E E Σ( ( r ( I = E the equality Σ( ( r ( statement b heorem or [5] and the set of and I the following statements are true: Σ that satisfy the condition I = E is fulfilled for the finite set (see ([] statement b heorem ([] By the symbol ( Σ 8 we denote the class of all semilattices of unions whose every element is = 7 5 where ( \ \ \ \ \ 5 \ \ \ \ \ isomorphic to an semilattice of form { he semilattice satisfying the conditions ( is shown in Figure Let ( { P P P P P P P P family sets where P 0 P P P P P 5 P P 7 are pairwise disjoint subsets of the set and C = is a 0 5 7

3 IJISE - International Journal of Innovative Science Engineering & echnology Vol Issue January 0 = P0 P P P P P5 P P 7 ϕ 5 7 ISSN is a mapping of the semilattice onto the family sets C( hen for the formal equalities of the semilattice we have a form: 5 7 Fig iagram of the Semilattice = P0 P P P P P5 P P7 = P0 P P P P5 P P7 = P0 P P P P5 P P7 = P0 P P P5 P P7 = P0 P P5 P P7 5 = P0 P P P P P7 = P0 P5 P7 = P P P 7 0 here the elements P P P P 5 are basis sources the element P0 P P P 7 is sources of completenes of the semilattice herefore and δ = (see ([] Item ([] Item or [] Now assume that Σ ( 8 Q { Q { Q { { 5 Q { We introduce the following notation: = where (see diagram in figure = where and (see diagram in figure = where and (see diagram in figure Q = where and (see diagram in figure 5 = where and \ \ (see diagram 5 in figure Q = { where { 7 { in figure 7 Q7 = { 7 in figure \ \ (see diagram where and \ \ (see diagram 8 Q8 = { where { 7 { 5 { \ \ ( \ \( and 9 Q { 0 Q { ( (see diagram 8 in figure 9 = where \ \ and = (see diagram 9 in figure 0 = where \ \ and = (see diagram 0 in figure Q = { 7 where { Q = { 7 where 7 Q = { where ( \ ( and = (see diagram in figure Q = { where ( and = (see diagram in figure and 7 = (see diagram in figure = (see diagram in figure ( \ \ \

4 IJISE - International Journal of Innovative Science Engineering & echnology Vol Issue January 0 5 Q5 = { where { 7 = ( \ ( diagram 5 in figure \ Q = { \ where 5 = (see diagram in figure Fig iagrams of all I subsemilattice of the semilattice { ISSN \ and = (see enote by the symbol ( Qi ( i to Q i Assume that ( Q i and denote by the symbol ( semigroup B ( for which the semilattices V ( and Q i are mutually isomorphic and V ( efinition Let the symbol I ( Let further ( and ϑ ( ( ϑ = the set of all I -subsemilattices of the semilattice isomorphic I the set of all idempotent elements of the Qi = denote the set of all I -subsemilattices of the semilattice I I I It is assumed that I if and only if there exists some complete isomorphism ϕ between the semilattices and One can easily verify that the binary relation ϑ I is an equivalence relation on the set I ( Let be an I -subsemilattices of the semilattice By I( we denoted the set of all idempotent elements of the semigroup B ( and I ( Q = I( where i = i 5 Qi ϑi Results = Σ 8 Lemma Let { ( 7 5 and 7 hen the following sets exhibit all I subsemilattices of the considered semilattice : { { { { { { 5 { { 7 (see diagram of the figure { { { { { { { { { { { 5 { 5 { { { { { { { (see diagram of the figure { 7 5 { 7 5 { 7 { 7 { 7 { 7 { 7 { { { { { { { { 5 { { { (see diagram of the figure 5

5 IJISE - International Journal of Innovative Science Engineering & echnology Vol Issue January 0 { 7 5 { 7 { 7 { { { (see diagram of the figure { { { { { { { (see diagram 5 of the figure 7 (see diagram of the figure 7 5 (see diagram 7 of the figure (see diagram 8 of the figure { { 7 { { 8 { 7 5 { ISSN Proof he statements - immediately follows from the heorems in [] the statements 5-7 immediately follows from the heorems in [] and the statement 8 immediately follows from the heorems 7 in [] he Lemma is proved = Σ 8 then the following equalities are true: Lemma If { ( I( Q = I( Q ( 7 5 \ \ = \ \ \ \ I( Q ( ( = \ \ \ \ \ \ I( Q ( ( ( = 5 ( ( \ \ \( ( I Q5 = \ = 5 \ ( \ ( \ 7 = ( 5 5 ( ( \ ( \ \ \ \ \ ( ( I Q \ \ 7 I( Q ( ( \ \ \( \( \ 8 ( ( ( ( I Q = 8 Proof: he statements - immediately follows from the Corollary 5 in [] the statements 5-7 immediately follows from the heorems the statement 8 immediately follows from the heorems 7 he Lemma is proved = Σ 8 B he binary heorem Let { ( 7 5 relation is an idempotent relation of the semigroup B ( following conditions: = where Y = where ( Y ( Y ( Y ( Y ( Y Y Y { 7 = where conditions: Y Y Y Y Y and ( iff the binary relation satisfies one of the and satisfies the conditions: Y Y Y Y { and satisfies the

6 IJISE - International Journal of Innovative Science Engineering & echnology Vol Issue January 0 ( Y ( Y ( Y ( Y0 ISSN = where { Y Y Y Y0 and satisfies the conditions: Y Y Y Y Y Y Y Y Y 0 5 ( Y ( Y ( Y ( Y ( = where \ \ Y Y Y { and satisfies the conditions: Y Y Y Y Y Y = where { { ( Y ( Y ( Y ( Y ( Y 0 \ \ Y Y Y Y { Y Y Y Y Y Y Y Y 7 and satisfies the conditions Y Y Y Y 7 ( Y ( Y ( Y ( Y ( ( Y0 = where \ \ Y Y Y Y Y Y Y { 0 0 and satisfies the conditions Y Y 8 ( Y ( Y ( Y ( Y ( ( Y ( Y0 Y Y = where { { { \ \ { 7 ( \ ( \ Y Y Y Y and satisfies the conditions Y Y Y Y Y Y Y Y Y Y Proof: In this case when 7 by Lemma we know that diagrams -8 given in Fig exhibit all diagrams of I subsemilattices of the semilattices a quasinormal representation of idempotent elements of the semigroup B ( 5 which are defined by these I semilattices may have one of the forms listed above he statements - immediately follows from the Corollary in [] the statements 5-7 immediately follows from the Corollary in [] and the statement 8 immediately follows from the heorems 7 he heorem is proved = Σ 8 If is a finite set then the number I ( Q Lemma Let { ( 7 5 ( I Q = 8 Proof: By the definition of the semilattice we have 7 Qϑ I = {{ 7 { { 5 { { { { { If the equalities { 7 { { 5 { 5 { { 7 { 8 { are fulfilled then I ( Q I( obtain ( = = = = = = = = 8 = (see efinition From this equality and statement of Lemma we I Q = = 8 he Lemma is proved Lemma Let = { Σ ( 8 I ( Q i If is a finite set then the number 7

7 IJISE - International Journal of Innovative Science Engineering & echnology Vol Issue January 0 I \ \ \ \ \ \ \ \ ( Q ( ( ( ( ( ( ( ( = \ 5 \ \ \ \ 7 \ \ \ ( \ \ \ 7 \ ( \ ( \ ( ( \ ( \ ( \ ( \ \ \ 7 \ 5\ 7 \ 5 ( ( ( \ \ \ 5 \ \ 7 \ \ Proof: By the definition of the semilattice we have Qθ = { { { { { { { { { { I 5 7 { 7 { { 5 { { 7 { { { 7 { 7 5 If the equalities = = = = = are fulfilled then { { { { { { { { { { 5 5 = 7 = 7 8 = 9 = 0 = = = = = = { { { { { { { { { = 7 = 8 = 7 9 = ( ( I Q = I i ISSN (see efinition From this equality and the statement of Lemma we obtain the validity of Lemma he Lemma is proved = Σ 8 If is a finite set then the number I ( Q Lemma 5 Let { ( I \ \ \ \ \ \ ( ( ( ( ( \ \ Q \ \ 5 ( ( ( ( \ \ \ \ \ \ \ + ( ( + ( ( \ \ ( ( ( ( \ 7 \ 7 ( ( ( ( \ \ 7 \ \ 5\ ( ( + ( ( = + + \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ ( ( ( ( \ \ 7 ( ( ( ( \ 7 \ \ \ ( ( ( ( \ + \ \ \ + + \ \ \ + \ \ \ \ \ \ \ Proof: By the definition of the semilattice we have Qθ I = {{ { { { 5 { { If the equalities { { { 7 { 7 { 7 { 7 5 { { { { 7 { 7 { 7 5 8

8 are fulfilled then IJISE - International Journal of Innovative Science Engineering & echnology Vol Issue January 0 ISSN = { = { = { = { 5 5 = { = { 7 = { 8 = { 9 = { 7 0 = { 7 = { 7 = { 7 5 = { = { 5 = { = = = { { { ( ( I Q = I i (see efinition From this equality and the statement of Lemma we obtain the validity of Lemma 5 he Lemma is proved = Σ 8 If is a finite set then the number I ( Q Lemma Let { ( I \ \ \ 5\ 7 \ 5 \ 5 ( Q ( ( ( \ 7 \ \ ( ( ( \ 7 \ \ ( ( ( \ \ \ + ( ( ( \ \ \ ( ( ( \ \ \ + ( ( ( = + \ \ \ + + \ \ \ + + \ \ \ + \ \ \ + + \ \ \ Proof: By the definition of the semilattice we have Qθ I = If the equalities are fulfilled then {{ { { { { { = = = { { { { { { = = = 5 ( ( I Q = I i (see efinition From this equality and the statement of Lemma we obtain the validity of Lemma he Lemma is proved = Σ 8 If is a finite set then the number Lemma 7 Let { ( I ( Q 5 I \ 5\ \ 5 ( Q5 ( ( ( ( \ ( ( ( ( \ \ \ + ( ( \ \ \ = + + 5\ \ 5 \ \ \ Proof By the definition of the semilattice we have Qθ I = {{ { { { { 7 5 { {

9 IJISE - International Journal of Innovative Science Engineering & echnology Vol Issue January 0 If the equalities = 7 = = = 7 5 = = = are fulfilled then { { { { { { { ( ( I Q5 = I i 7 ISSN (see efinition From this equality and the statement 5 of Lemma we obtain the validity of Lemma 7 he Lemma is proved = Σ 8 If is a finite set then the number I ( Q Lemma 8 Let { ( ( ( ( ( ( \ 7 \ \ \ \ \ \ I Q = 5 + \ ( \ \ \ \ \ \ + ( ( ( 5 Proof: By the definition of the semilattice we have Qθ I = {{ 7 { If the equalities = { 7 = { are fulfilled then ( = ( + ( I Q I I (see efinition From this equality and the statement of Lemma we obtain the validity of Lemma 8 he Lemma is proved = Σ 8 If is a finite set then the number I ( Q 7 Lemma 9 Let { ( 7 5 I 7 \ \ \ \ 5 5\ ( Q7 ( ( ( \ \ + ( ( ( = \ \ \ 5 5 Proof: By the definition of the semilattice we have Q7θ I = {{ 7 5 { If the equalities = { 7 5 = { are fulfilled then ( = ( + ( I Q I I 7 (see efinition From this equality and the statement 7 of Lemma we obtain the validity of Lemma 9 he Lemma is proved = Σ 8 If is a finite set then the Lemma 0 Let { ( number I ( Q \ \ \ \ \ I ( Q8 = ( + \ 5\ \ 5 \ \ + ( ( ( Proof: By the definition of the semilattice we have Q8θ I = {{ 7 5 { If the equalities = { 7 5 = { are fulfilled then ( = ( + ( I Q I I 8 (see efinition From this equality and the statement 8 of Lemma we obtain the validity of Lemma 0 70

10 IJISE - International Journal of Innovative Science Engineering & echnology Vol Issue January 0 he Lemma is proved Let us assume that 8 ( k = I Qi heorem Let = { Σ ( of all idempotent elements of the semigroup ( B then I = k Proof: his heorem immediately follows from the heorem he heorem is proved 7 Example Let = { 5 { { { { { ISSN If is a finite set and I is a set P = P = P = P = P = 5 P = P = P = hen = { 5 = { 5 = { 5 = { 5 = { 5 5 { = { 5 7 = { and = {{ { 5 { { 5 { 5 { 5 { 5 { 5 herefore we have that following equality and inequality is valid: { { 7 = 5 = where I ( Q = 8 I ( Q = 7 I ( Q = 5 I ( Q = I ( Q 5 = 7 I ( Q = ( ( I Q 8 = I = I Q 7 = Reference [] Ya iasamidze Sh Makharadze Complete Semigroups of binary relations Monograph Kriter urkey 0-50 pp [] Ya iasamidze Sh Makharadze Complete Semigroups of binary relations Monograph M Sputnik p (Russian [] Ya I iasamidze Complete Semigroups of Binary Relations Journal of Mathematical Sciences Plenum Publ Cor New York Vol 7 No [] iasamidze Ya Makharadze Sh Rokva N On I semilattices of union Bull Georg Nation Acad Sci [5] iasamidze Ya Makharadze Sh iasamidze Il Idempotents and regular elements of complete semigroups of binary relations Journal of Mathematical Sciences Plenum Publ Cor New York