Complete Semigroups of Binary Relations Defined by Semilattices of the Class Σ X,10

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1 Applied Mathematics Published Online February 05 in SciRes. Complete Semigroups of Binary Relations efined by Semilattices of the Class Σ X 0 ( Shota Makharadze Neşet Aydın Ali Erdoğan Shota Rustavelli University Batum Georgia Çanakkale Onsekiz Mart University Çanakkale urkey Hacettepe University Ankara urkey shota5@mail.ru neseta@comu.edu.tr alier@hacettepe.edu.tr Received 0 January 05; accepted 8 January 05; published 4 February 05 Copyright 05 by authors and Scientific Research Publishing Inc. his work is licensed under the Creative Commons Attribution International License (CC BY. Abstract In this paper we give a full description of idempotent elements of the semigroup B X ( which are defined by semilattices of the class Σ (X 0. For the case where X is a finite set we derive formulas by means of which we can calculate the numbers of idempotent elements of the respective semigroup. Keywords Semilattice Semigroup Binary Relation. Introduction Let X be an arbitrary nonempty set be an X-semilattice of unions i.e. such a nonempty set of subsets of the set X that is closed with respect to the set-theoretic operations of unification of elements from f be an arbitrary mapping of the set X in the set. o each such a mapping f we put into correspondence a binary relation on the set X that satisfies the condition f f ({ x} f ( x = x X he set of all such f f : X X BX is a semigroup with respect to the operation of multiplication of binary relations which is called a complete semigroup of binary relations defined by an X-semilattice of unions. ( is denoted by B (. It is easy to prove that ( How to cite this paper: Makharadze S. Aydın N. and Erdoğan A. (05 Complete Semigroups of Binary Relations efined by Semilattices of the Class Σ ( X 0. Applied Mathematics

2 Recall that we denote by an empty binary relation or empty subset of the set X. he condition ( xy will be written in the form xy. Further let x y X Y X BX ( = and t. hen by symbols we denoted the following sets: X X y = { x X yx} Y = y = { YY X} X = \{ } y Y V( = { Y Y } = { } = { } t = { t } l ( = ( \. By symbol ( Λ is denoted an exact lower bound of the set ' in the semilattice. efinition. We say that the complete X-semilattice of unions is an XI-semilattice of unions if it satisfies the following two conditions: a Λ( t for any t ; b = Λ ( t for any nonempty element of the semilattice. t efinition. We say that a nonempty element is a nonlimiting element of the set ' if \ l( and a nonempty element is a limiting element of the set ' if \ l( =. efinition. Let BX ( V ( X Y = { y X y = }. A representation of a binary relation of the form = Y V( X ( is called quasinormal. Note that if = Y V( X ( is a quasinormal representation of the binary relation then the following conditions are true: X = Y V( X ; Y Y = for V ( X and. Let ( X m denote the class of all complete X-semilattices of unions where every element is isomorphic n to a fixed semilattice. he following heorems are well know (see [] and []. heorem 4. Let X be a finite set; δ and q be respectively the number of basic sources and the number of all automorphisms of the semilattice. If X n δ where C k = j! = and ( Σ n X m = s then δ δ ( m δ p ( δ! (( δ! (! (! m p C C p i s = q p= δ i= i p i (see heorem.5. []. p i p n j ( k! ( j k! heorem 5. Let be a complete X-semilattice of unions. he semigroup B ( X possesses right unit iff is an XI-semilattice of unions (see heorem 6.. []. heorem 6. Let X 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 V is an idempotent element of this semigroup iff ( ( V is complete XI-semilattice of unions; Y ( ; c Y (see heorem 6.. []. and I denote respectively the complete X-semilattice of unions the set and the set of and I the following statements are true: a ( b for any ( for any nonlimiting element of the set ( ( r heorem 7. Let Σ ( E X ( of all XI-subsemilatices of the semilattice the set of all right units of the semigroup BX ( ( r all idempotents of the semigroup BX (. hen for the sets E X ( if and Σ ( = { ( } then ( r a ( r E ( E ( = for any elements and of the set ( X ; r b I = E X X ( c the equality ( ( r ( I = E is fulfilled for the finite set X. X Σ that satisfy the condition 75

3 if then ( r a ( r E ( X EX ( ; r b I = E X = for any elements and of the set ( ( c the equality I ( ( E r X ( Corollary. Let Y { y y yk } j the number ( k can be calculated by the formula s( k j j ( j Σ that satisfy the condition = is fulfilled for the finite set X (see heorem 6.. []. = be some sets where k and j. hen s k j of all possible mappings of the set Y into any such subset j of the set j that j j = k (see Corollary.8. []. = and { j}. Idempotent Elements of the Semigroups BX ( Class Σ ( X 0 Let X and ( X efined by Semilattices of the Σ 0 be respectively an arbitrary nonempty set and a class X-semilattices of unions where each element is isomorphic to some X-semilattice of unions = { } that satisfies the conditions: \ \ 8 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ = = = 4 = = = = = = = = = 7 = 8 = 8 = = = = = = = An X-semilattice that satisfies conditions ( is shown in Figure. C = P P P P P P P P P P be a family of sets where P 0 P P P P 4 P 5 P 6 P 7 P 8 P Let ( { } ( Figure. iagram of. 76

4 are pairwise disjoint subsets of the set X and ping of the semilattice onto the family sets ( have a form: ϕ S. Makharadze et al. = be a map- P0 P P P P4 P5 P6 P7 P8 P C. hen for the formal equalities of the semilattice we = P0 P P P P4 P5 P6 P7 P8 P = P0 P P P4 P5 P6 P7 P8 P = P0 P P P4 P5 P6 P7 P8 P = P0 P P P4 P5 P6 P7 P8 P 4 = P0 P P P5 P6 P7 P8 P 5 = P0 P P P4 P6 P7 P8 P 6 = P0 P4 P5 P7 P8 P 7 = P0 P P P4 P5 P6 P8 P 8 = P0 P P P4 P5 P6 P7 P = P. 0 Here the elements P P P P 4 P 5 P 6 P 7 P 8 are basis sources the elements P 0 P 6 P are sources of completeness of the semilattice. herefore X 7 and δ = 7 (see []. Lemma. Let ( X0 Σ ( X0 = s and X δ 7. If X is a finite set then n n n n n n n s = (( Proof. In this case we have: m = 0 δ = 7. Notice that an X-semilattice given in Figure has eight automorphims. By heorem. it follows that where C k j j! = k! j k! ( p i 0 p p 7 7 n ( C Cp ( 7! ( ( p 7! i p= 7 i= ( i ( p i s = 8!! and that Example 8. Let n = hen: Lemma. Let ( X n n n n n n n s = (( X ( B =. 0. hen the following sets are all proper subsemilattices of the semilattice { = } : { } { 8} { 7} { 6} { 5} { 4} { } { } { } { } (see diagram of the Figure ; { 8} { 7} { 6} { 5} { 4} { } { } { } { } { 8 } { 8 } { 7 } { 7 } { 6 } { 6 } { 6 } { 6 } { 5 } { 5 } { 4 } { 4 } { } { } { } (see diagram of the Figure ; { } { } { } { } { } { } { } { 6 } { 5 } { 5 } { 4 } { 4 } { } { } ( 77

5 { } { 8 } { 7 } { 6 } { 6 } { 6 } { 5 } { 4 } (see diagram of the Figure ; { } { } { } { } { } { 7 } { 8 } (see diagram 4 of the Figure ; 5 { } { } { } { } { } { } ; { } { } { } { } { } { } { } { } { } { } { } { } { } { } { } { 6 } { 6 } { 6 } (see diagram 5 of the Figure ; { } { } { } { } { 8 6 } { 8 7 } (see diagram 6 of the Figure ; 7 { 6 } { 6 } { 6 } (see diagram 7 of the Figure ; { } { } { } { } { 6 5 } { 6 5 } { 6 4 } { 6 4 } (see diagram 8 of the Figure ; { 8 7 } { 8 6 } { 8 6 } { 8 5 } { 8 4 } { 8 } { 8 } { 7 6 } { 7 5 } { 7 4 } { 7 } { 7 } { 6 5 } { 6 4 } { 5 4 } { 5 } { 5 } { 4 } { 4 } { } { } { } (see diagram of the Figure ; 0 { 8 6 } { 8 7 } { 7 6 } { 5 4 } { 6 4 } { 6 5 } (see diagram 0 of the Figure ; { } { } { } { } { 7 6 } { 7 6 } { 8 6 } { 8 6 } (see diagram of the Figure ; { } { } { 8 } { } { 4 } { 5 } { 7 } { 7 4 } { 7 5 } { 8 4 } { 8 5 } (see diagram of the Figure ; { 5 } { 5 } { 7 } { 7 } { 7 4 } { 7 4 } { } { } { } { } { } { } { 7 5 } { 8 4 } { 8 5 } { 8 5 } (see diagram of the Figure ; 4 { } { } { 6 } { } { 4 } 78

6 { 5 } { 7 } { 7 4 } { 7 4 } { 7 5 } { 8 } { 8 4 } { 8 5 } (see diagram 4 of the Figure ; { 7 } { 7 4 } { 7 5 } { 7 5 } { 8 } 5 { } { } { } { } { } { 8 } { 8 4 } { 8 4 } { 8 5 } { 8 5 } (see diagram 5 of the Figure ; { } { } { } { } { } { 8 7 } { } { } (see diagram 6 of the Figure ; { 7 5 } { 7 4 } { 8 } { 8 5 } { 8 5 } 7 { } { } { } { } { } { 8 4 } { 8 4 } (see diagram 7 of the Figure ; 8 { 7 4 } { 7 5 } { 8 5 } { 8 4 } (see diagram 8 of the Figure ;. { } { } (see diagram of the Figure ; { } { } { } { } { 8 7 } { 5 4 } (see diagram 0 of the Figure ; { } { } { } { } (see diagram of the Figure ; { } { } { } { } { } { 5 4 } { 8 7 } { } (see diagram of the Figure ; { } { } (see diagram of the Figure ; { 8 } { 7 5 } { 7 5 } { 7 4 } 4 { } { } { } { } { 7 4 } { 7 } { 5 } { 4 } (see diagram 4 of the Figure ; 5 { 8 5 } { 8 4 } { 7 5 } { 7 4 } (see diagram 5 of the Figure ; 6 { 6 } (see diagram 6 of the Figure ; 7

7 7 { } { } { } { } (see diagram 7 of the Figure ; { } { } { } { } { } (see diagram 8 of the Figure ; { 8 6 } { 7 6 } { 6 5 } { 6 4 } (see diagram of the Figure ; 0 { 8 4 } { 7 5 } { 7 4 } (see diagram 0 of the Figure ; { } (see diagram of the Figure ; { } { } (see diagram of the Figure ; { } { } { } { } (see diagram of the Figure ; { } { } { } { } (see diagram 4 of the Figure ; { } { } { } { 8 6 } { 8 6 } (see diagram 5 of the Figure ; 6 { } { } { } (see diagram 6 of the Figure ; { } { } { } { 8 5 } (see diagram 7 of the Figure ; 8 { } { } { } (see diagram 8 of the Figure ; { } (see diagram of the Figure ; { } { } { } { } { 5 4 } (see diagram 40 of the Figure ; 4 { } { } (see diagram 4 of the Figure ; 80

8 4 { } (see diagram 4 of the Figure ; 4 { } { } { } (see diagram 4 of the Figure ; 44 { } (see diagram 44 of the Figure ; 45 { } (see diagram 45 of the Figure ; 46 { } { } (see diagram 46 of the Figure ; { } { } { } { } (see diagram 47 of the Figure ; { } { } { } { } (see diagram 48 of the Figure ; S. Makharadze et al. Figure. iagram of all subsemilattices of. 8

9 4 { } (see diagram 4 of the Figure ; { } { } { } { } (see diagram 50 of the Figure ; 5 { } (see diagram 5 of the Figure ; 5 { } (see diagram 5 of the Figure ; iagrams of subsemilattices of the semilattice. Lemma. Let ( X0. hen the following sets are all XI-subsemi-lattices of the given semilattice : { } { 8} { 7} { 6} { 5} { 4} { } { } { } { } (see diagram of the Figure ; { 8 } { 7 } { 7 } { 6 } { 6 } { 6 } { 6 } { 5 } { 5 } { 4 } { } { } { } { } { } { } { } { } { } { } { 4 } { } { } { } (see diagram of the Figure ; { 8 } { 7 } { 6 } { 5 } { 4 } { } { } { } { } { } { } { } { } { } { } { 8 } { 7 } { 6 } { 6 } { 6 } { 5 } { 4 } (see diagram of the Figure ; { } { } { } { } { } { 7 } { 8 } (see diagram 4 of the Figure ; 5 { } { } { } { } { } { } { } { } { } { } { } { } { } { } { } { } { } { } { } { } { } { 6 } { 6 } { 6 } (see diagram 5 of the Figure ; { } { } { } { } { 8 6 } { 8 7 } (see diagram 6 of the Figure ; 7 { 6 } { 6 } { 6 } (see diagram 7 of the Figure ; 8

10 8 { 8 6 } { 8 6 } { 7 6 } { 7 6 } { 6 5 } { 6 5 } { 6 4 } { 6 4 } (see diagram 8 of the Figure ; S. Makharadze et al. Proof. It is well know (see [] that the semilattices to 8 which are given by lemma are always XI-semilattices. he semilattices and 0 which are given by Lemma { 8 7 } { 8 6 } { 8 6 } { 8 5 } { 8 4 } { 8 } { 8 } { 7 6 } { 7 5 } { 7 4 } { 7 } { 7 } { 6 5 } { 6 4 } { 5 4 } { 5 } { 5 } { 4 } { 4 } { }. { } { } (see diagram of the Figure ; { 8 6 } { 8 7 } { 7 6 } { 5 4 } { 6 4 } { 6 5 } (see diagram 0 of the Figure ; are XI-semilattices iff the intersection of minimal elements of the given semilattices is empty set. From the formal equalities ( of the given semilattice we have = P P P P P P P P P P P P P P P P = P P P P P P P P P P P P P P = P P P P P P P P P P P P P P P P = P P P P P P P P P P P P P P P P = P P P P P P P P P P P P P P P P P = P P P P P P P P P P P P P P P P P = P P P P P P P P P P P P P P = P P P P P P P P P P P P P P P P = P P P P P P P P P P P P P P P P = P P P P P P P P P P P P P P P P P = P P P P P P P P P P P P P P P P P = P P P P P P P P P P P P P P = P P P P P P P P P P P P P P = P P P P P P P P P P P P P P P P = P P P P P P P P P P P P P P P P P = P P P P P P P P P P P P P P P P P = P P P P P P P P P P P P P P P P P

11 = P P P P P P P P P P P P P P P P P = P P P P P P P P P P P P P P P P P P = P P P P P P P P P P P P P P P P P P = P P P P P P P P P P P P P P P P P P From the equalities given above it follows that the semilattices and 0 are not XI-semilattices. he semilattices { 6 5 } { 6 5 } { 6 4 } { 6 4 }. { 7 6 } { 7 6 } { 8 6 } { 8 6 } (see diagram -8 of the Figure ; are not XI-semilattice since we have the following inequalities he semilattices to 5 are never XI-semilattices. We prove that the semilattice diagram 5 of the Figure Q = and is not an XI-semilattice (see Figure 4. Indeed let { } C( Q = { P P P P P P P P P } be a family of sets where P 0 P P P P 4 P 5 P 6 P 7 P 8 are pairwise disjoint subsets of the set X. Let = P 0 P P P P 4 P 5 P 6 P 7 P 8 ϕ be a mapping of the semilattice Q onto the family of sets C( Q. hen for the formal equalities of the semilattice Q we have a form: Figure. iagram of all subsemilattices which are isomorphic to in Figure. Figure 4. iagram of subsemilattice 5 in Figure. 84

12 0 = P 0 P P P P 4 P 5 P 6 P 7 P 8 = P 0 P P P 4 P 5 P 6 P 7 P 8 = P 0 P P P 4 P 5 P 6 P 7 P 8 = P 0 P P 4 P 5 P 6 P 7 P 8 4 = P 0 P P P 5 P 6 P 7 P 8 5 = P 0 P P 4 P 6 P 7 P 8 6 = P 0 P P P 4 P 5 P 7 P 8 ' 7 = P 0 P P P 4 P 5 P 6 P8 = P. 8 0 Here the elements P P P P 4 P 6 P 7 are basis sources the elements P 0 P 5 P 8 are sources of completeness of the semilattice. herefore X 6 and δ = 7 (see []. hen of the formal equalities we have: Q if t P 0 { 7 6 0} if t P { 4 0} if t P { } if t P Q { } if 4 t = t P { } if t P 5 { } if t P 6 { } if t P 7 { } if t P 8. We have that = { } and ( QQ 8 if t P 0 8 if t P 8 if t P 8 if t P Λ ( QQ t = 8 if t P 4 8 if t P 5 8 if t P 6 8 if t P 7 8 if t P 8. Q 8 Λ t Q for any t Q. But elements are not union of some elements of the set Q. herefore from the efinition it follows that Q is not an XI-semilattice of unions. Statements to 5 can be proved analogously. We denoted the following semitattices by symbols: = where (see diagram of the Figure 5; a Q { } b Q { } c Q { } = where and (see diagram of the Figure 5; = where and (see diagram of the Figure 5; d Q 4 { } = where and (see diagram 4 of the Figure 5; e Q5 = { } where \ \ (see diagram 5 of the Figure 5; f Q = 6 { } where \ \ (see diagram 6 of the Figure 5; g Q7 = { 6 } where 6 6 \ \ = (see diagram 7 of the Figure 5; ( 85

13 h Q8 { } Figure 5. iagram of all XI-subsemilattices of. = where \ \ ( \ \ ( (see diagram 8 of the Figure 5; Note that the semilattices in Figure 5 are all XI-semilattices (see [] and Lemma... efinition. Let us assume that by the symbol Σ XI ( X denote a set of all XI-subsemilatices of X-semilatices of unions that every element of this set contains an empty set if or denotes a set of all XI-subsemilatices of. Further let XI ( X and ϑ XI Σ XI ( X Σ XI ( X. It is assumed that ϑxi iff there exists some complete isomorphism ϕ between the semilatices and. One can easily verify that the binary relation ϑ XI is an equivalence relation on the set Σ XI ( X. By the symbol Qϑ i XI denote the ϑxi -equivalence class of the set Σ XI ( X where every element is isomorphic to the X-semilattice Q i ( i = 8. Let ' be an XI-subsemilattice of the semilattice. By I( we denoted the set of all right units of the semigroup BX ( and I ( Qi = I( Qi ϑxi where i = 8. Lemma 4. If X is a finite set then the following equalities hold a I( Q = b I( Q ( \ = c I( Q \ \ \ = \ \ \ \ \ d I( Q ( 4 = 4 e ( ( \ \ \( ( I Q5 = 4 X \ ( \ ( 6 = ( \ \ f I( Q \ ( 6 \ \ \ \ g I( Q ( \ \ \( \( \ h ( 8 ( I Q = 6 X 6 \ X \ 7 = 5 Proof. his lemma immediately follows from heorem... and.7. of the []. heorem 0. Let ( X0 and BX (. Binary relation is an idempotent relation of the semmigroup BX ( iff binary relation satisfies only one conditions of the following conditions: a = X where ; b ( Y = ( Y where Y Y { } and satisfies the conditions: Y Y ; = Y Y Y where Y Y Y { } and sa- c ( ( ( tisfies the conditions: Y Y Y Y Y ; 86

14 d ( Y ( Y ( Y ( Y0 { } = where Y Y 0 and satisfies the conditions: Y Y 0 ; e ( Y ( Y ( Y ( Y ( Y Y Y Y Y Y Y S. Makharadze et al. = where \ \ Y Y Y and satisfies the conditions: Y Y Y Y Y Y ; f = ( Y ( Y ( Y Y ( Y0 where { } { } Y Y \ \ Y Y Y 0 and satisfies the conditions: Y Y Y Y Y Y Y 0 ; = Y Y 6 6 Y Y Y 0 where 6 6 g ( \ \ Y6 Y Y and satisfies the conditions: Y Y Y6 6 Y Y6 Y Y Y6 Y Y 6 6 Y Y ; h = ( Y ( Y ( Y ( Y ( ( Y ( Y0 where \ \ ( \ \ ( Y Y Y { } and satisfies the condi tions: Y Y Y Y Y Y Y Y Y Y. Proof. By Lemma we know that to 8 are an XI-semilattices. We prove only statement g. Indeed if = { } ( Y ( Y 6 6 ( Y ( Y ( Y 0 = 6 = 6 is a generating set of the 6. hen the following equalities hold = = where Y Y Y { } then it is easy to see that the set ( { } semilattice { } ( { } ( { } 6 6 ( = { } ( = { }. By statement a of the heorem 6.. (see [] we have: 6 6 Y Y Y Y Y Y Y Y Y Further one can see that the equalities are true: ( ( ( { } ( ( ( ( l = \ = \ l = \ ( ( ( { } ( ( ( ( ( ( { } ( ( l = \ = \ l = \ 6 6 l = \ = \ l = \ 6 6 We have the elements 6 ' are nonlimiting elements of the sets ( 6 ( ( respectively. By statement b of the heorem 6.. [] it follows that the conditions Y 6 6 Y Y hold. herefore the statement g is proved. Rest of statements can be proved analogously. Lemma 5. Let ( X0 and. If X is a finite set then the number I ( Q may be calculated by the formula I ( Q = 0. Lemma 6. Let ( X0 and. If X is a finite set then the number I ( Q may be calculated by formula 8\ \ 7\ 6\ 5\ 8 \ 7 \ X X X 6 5 I ( Q = 4\ \ \ 8 \ 7 \ \ \ 6 \ \ 6 \ 5 \ 4 X 4 \ \ \ 8 \ 7 \ 6 \ 5 \ 4 \ \ \. 87

15 Lemma 7. Let ( X may be calculated by formula 0 and I Q. If X is a finite set then the number ( 8\ \ 8 \ 8 \ 7\ X \ 7 \ 7 I ( Q = Lemma 8. Let ( X 6\ \ 6 \ 6 5\ \ 5 \ 5 4\ \ 4 \ 4 \ \ \ \ \ \ \ \ \ 8 \ \ 8 \ 8 7\ \ 7 \ 7 6\ \ 6 \ 6 6\ \ 6 \ 6 6\ \ 6 \ 6 \ 5\ \ 5 \ X 5 0 and calculated by formula 0 and calculated by formula Lemma. Let ( X 4\ \ 4 \ 4 \ 8 \ \ \ X \ \ 7 \ \ \ 6 \ \ X \ 6 \ \ \ 6 \ \ \ 5 \ \ \ \ 4 \ \ \ X X. I I Q 4. If X is a finite set then the number ( \ \ 8\ ( Q4 = ( \ 8 ( \ 8 ( 4 ( 7\ ( \ 7 \ 7 ( ( 6\ ( \ 6 \ 6 ( 6\ ( \ 6 ( \ 6 ( ( 6\ ( \ 6 \ 6 ( ( 5\ ( \ 5 \ 5 ( 4\ \ 4 \ 4 \ \ ( ( ( 4 \ \ 4 \ \ 4 \ \ 4 \ \ 4 4 \ \ I Q 5. If X is a finite set then the number ( may be may be 88

16 I Lemma 0. Let ( X lated by formula 5\ 4 ( Q5 = ( 4\ 5 ( 4 6\ 4 ( 4\ 6 ( 6\ 5 ( 5\ 6 ( 4 7\ 6 ( 6\ 7 ( 8\ 6 ( 6\ 8 ( 4 8\ 7 ( 7\ 8 ( 8\ 4 4\ 8 8\ 5 5\ 8 ( ( 4 ( ( 7\ \ 7 7\ 4 4\ 7 ( ( 4 ( ( 7\ 5 5\ 7 8\ \ 8 ( ( 4 ( ( 8 \ \ 8 4\ \ 4 ( ( 4 ( ( 4\ \ 4 5\ \ 5 ( ( 4 ( ( 5\ \ 5 7\ \ 7 ( ( 4 ( ( \ \ X \ ( ( 4 ( \ ( \ \ ( (. 0 and 0 and Lemma. Let ( X lated by formula I Lemma. Let ( X lated by formula \ I Q 6. If X is a finite set then the number ( ( I Q 6 = = 4 I Q 7. If X is a finite set then the number ( ( ( \ Q 7 = ( \ 6 ( \ ( \ \ \ 5 ( 6 \ ( \ 6 ( \ ( \ \ \ ( 6 \ ( \ 6 ( \ ( \ \ \ and I I Q 8. If X is a finite set then the number ( 8\ ( Q8 = ( 6\ 8 ( \ ( \ 6 ( 8\ ( 6\ 8 ( \ \ ( 7\ ( 6\ 7 ( \ \ 7\ ( 6\ 7 ( \ ( \ ( 5\ ( 6\ 5 ( \ \ \ \ ( 4\ 6\ 4 \ \ ( 4\ 6\ 4 \ \ ( ( ( \ \ 6 6 Figure 6 shows all XI-subsemilattices with six elements may be calcu- may be calcu- may be calcu- 8

17 heorem. Let ( X Figure 6. iagram of all subsemilattices which are isomorphic. 0. If X is a finite set and the semigroup BX (. hen I 8 I ( Q i= i Example. Let X = { 45678} =. { } { } { } { } { } { 5 } { 7 } { 8 }. P = 6 P = P = P = P = P = P = P = P = P = I is a set of all idempotent elements of hen = { 45678} = { 45678} = { 45678} = { 45678} 4 = { 5678} 5 = { 4678} 6 = { 45678} 7 = { 4568} 8 { 4567} = { 6}. = We have {{ } { } { } { } { } { 4678 }{ }{ 4568 }{ 4567 }{ 6}} Where I ( Q = 0 I ( Q = 6 I ( Q = 54 ( ( = I ( Q 6 = 4 I ( Q 7 = 0 ( I Q 5. Results Lemma. Let ( X I Q = I = = and I Q 4 = 4 0 and =. hen the following sets exhaust all subsemilattices of the semi- = which contains the empty set: lattice { } { } (see diagram of the Figure ; { } { } { } { } { } { } { } { } { } (see diagram of the Figure ; { 8 } { 7 } { 6 } { 5 } { 4 } { } { } { } { } { } { } { } { } { } { } (see diagram of the Figure ; 0

18 4 { 4 } { 5 } { 6 } { 6 } { 6 } { 7 } { 8 } (see diagram 4 of the Figure ; 5 { } { } { } { } { } { } S. Makharadze et al { } { } { } { } { } { } { } { } { } { } { } { } { } { } { } (see diagram 5 of the Figure ; { } { } { } { } { 8 6 } { 8 7 } (see diagram 6 of the Figure ; 7 { 6 } { 6 } { 6 } (see diagram 7 of the Figure ; { } { } { } { } { 6 5 } { 6 5 } { 6 4 } { 6 4 } (see diagram 8 of the Figure ; heorem. Let ( X0 of the semmigroup B ( BX. Binary relation is an idempotent relation X iff binary relation satisfies only one conditions of the following conditions: a = ; Y = Y where Y and satisfies the conditions: Y ; b c ( Y ( Y ( Y = and ( = where Y Y { } and satisfies the conditions: Y Y Y Y ; = Y Y Y Y 0 where Y { } d Y Y 0 and satisfies the conditions: Y Y Y Y Y Y Y Y 0 ; e = ( Y ( Y ( Y Y ( where \ \ Y { } ( Y and satisfies the conditions: Y Y Y Y f ( ( ( ( 0 Y { } 0 Y Y Y ; = Y Y Y ( Y ( Y where \ \ Y Y 0 and satisfies the conditions: Y Y Y Y Y Y. = Y Y 6 6 Y Y Y 0 where 6 6 g ( \ \ Y6 Y Y and satisfies the conditions: Y Y6 6 Y Y6 Y Y Y6 Y Y 6 6 Y Y ; = Y Y Y Y Y Y where = { } h ( ( ( ( ( ( ( 0 \ \ ( \ \ Y Y Y and satisfies the conditions: Y Y Y Y Y Y Y Y Y Y ; Lemma 4. Let ( X0 and =. If X is a finite set then I ( Q =. Lemma 5. Let ( X0 and =. If X is a finite set then the number I ( Q may be calcu- ( { }

19 lated by formula I ( Q = Lemma 6. Let ( X lated by formula and I Q =. If X is a finite set then the number ( 8 \ 8 \ 8 7 \ 7 \ 7 I ( Q = Lemma 7. Let ( X lated by formula 6 \ 6 \ 6 5 \ 5 \ 5 \ 4 \ 4 \ 4 X \ \ \ \ \ \ 8 \ 8 \ 8 7 \ 7 \ 7 6 \ 6 \ 6 6 \ 6 \ 6 6 \ 6 \ 6 4 \ 4 \ 4. 0 and Lemma 8. Let ( X I 5 \ 5 \ 5 I Q 4 =. If X is a finite set then the number ( X 4 ( Q4 = ( \ 4 ( \ 4 ( 4 \ \ \ 5 ( \ 5 ( \ 5 ( 4 6 ( \ 6 ( \ 6 ( 4 6 \ 6 \ 6 \ ( ( ( 4 ( 6 ( \ 6 \ 6 ( ( 7 ( \ 7 \ 7 ( ( 8 ( \ 8 \ 8 ( 0 and \ \ \ \ \ 4 \ \ 4 4 \ \ 4 4 \ \ 4 4. I Q 5 =. If X is a finite set then the number ( may be calcumay be calcu- may be calcu-

20 lated by formula Lemma. Let ( X lated by formula I 5\ 4 ( Q5 = ( 4\ 5 ( 4 6\ 4 ( 4\ 6 ( ( ( 4 ( ( ( ( 4 ( ( 8\ 4 4\ 8 X 8\ 5 ( ( 4 ( 5\ 8 ( 7\ \ 7 X 7\ 4 ( ( 4 ( 4\ 7 ( 7\ 5 5\ 7 X 8\ \ 8 ( ( 4 ( ( 8 \ \ 8 X 4\ ( ( 4 ( \ 4 ( 4\ \ 4 X 5\ ( ( 4 ( \ 5 ( 5\ \ 5 X 7\ \ 7 ( ( 4 ( ( \ \ X \ \ ( ( 4 ( ( \ \ ( (. 0 and Lemma 0. Let ( X lated by formula I 6\ 5 5\ 6 7\ 6 6\ 7 8\ 6 6\ 8 8\ 7 7\ 8 \ \ \ \ \ \ \ I Q 6 =. If X is a finite set then the number ( X 5\ 4 ( Q6 = ( 4\ 5 ( ( 5 \ 4 \ 5 \ ( 6\ 4 ( 4\ 6 ( ( 6\ 5 ( 5\ 6 ( 7\ 6 6\ 7 \ \ ( ( ( 5 4 ( 8\ 6 ( 6\ 8 ( ( 8\ 7 ( 7\ 8 ( 0 and Lemma. Let ( X I \ \ \ \ \ \ \ \ I Q 7 =. If X is a finite set then the number ( ( 6 \ 6 \ \ \ \ Q 7 = 5 ( 6 ( 6 \ \ \ \ \ ( 6 ( 6 \ \ \ \ \ and I Q 8 =. If X is a finite set then the number ( may be calcu- may be calcu- may be calcu-

21 lated by formula I 8\ 6\ 8 \ \ ( Q8 = ( ( ( 6 8\ ( 6\ 8 ( \ ( \ 6 7\ ( 6\ 7 ( \ ( \ 6 7\ ( 6\ 7 ( \ ( \ 6 5\ ( 6\ 5 ( \ ( \ 6 5\ ( 6\ 5 ( \ ( \ 6 4\ ( 6\ 4 ( \ ( \ 6 4 \ ( ( 6\ 4 ( \ \ 6. heorem 4. Let ( X0 =. If X is a finite set and the semigroup BX ( then 8 I = I ( Q i= i. Example 5. Let X = { 4567} { } { } { } { } { } { } { } I is a set of all idempotent elements of P = P = P = P = 4 P = 5 P = 6 P = 7 P = P = P = hen = { 4567} = { 4567} = { 4567} { 4567} 4 = { 567} 5 = { 467} 6 = { 4567} 7 = { 457} 8 { 456} We have = = = and =. {{ } { } { } { } { } { 467 }{ 4567 }{ 457 }{ 456 } } I Q 5 = =. Where I ( Q = I ( Q = I ( Q = 4 I ( Q 4 = 4 ( I ( Q 6 = 4 I ( Q 7 = 0 I( Q 8 = 8 I = 84. It was seen in ([4] heorem that if and β are regular elements of BX ( then V ( β XI-subsemilattice of. herefore β is regular elements of BX ( of BX ( is a subsemigroup of BX (. References is an. hat is the set of all regular elements [] iasamidze Ya. and Makharadze Sh. (0 Complete Semigroups of Binary Relations. Monograph. Kriter urkey 60 p. [] iasamidze Ya. and Makharadze Sh. (00 Complete Semigroups of Binary Relations. Monograph. M. Sputnik 657 p. (In Russian [] iasamidze Ya. Makharadze Sh. and iasamidze Il. (008 Idempotents and Regular Elements of Complete Semigroups of Binary Relations. Journal of Mathematical Sciences Plenum Publ. Cor. New York [4] iasamidze Ya. and Bakuridze Al. (to appear On Some Properties of Regular Elements of Complete Semigroups efined by Semilattices of the Class Σ 4 ( X 8. 4

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