Batumi, GEORGIA 2 Department of Mathematics. Hacettepe University Beytepe, Ankara, TURKEY 3 Department of Mathematics

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1 International Journal of Pure and Applied Mathematics Volume 93 No , ISSN: printed version; ISSN: on-line version url: doi: PAijpam.eu SOME REGULAR ELEMENTS, IDEMPOTENTS AND RIGHT UNITS OF COMPLETE SEMIGROUPS OF BINARY RELATIONS DEFINED BY SEMILATTICES OF THE CLASS LOWER INCOMPLETE NETS Yasha Diasamidze 1, Ali Erdogan 2, Neşet Aydın 3 1 Sh. Rustaveli State University Batumi, GEORGIA 2 Department of Mathematics Hacettepe University Beytepe, Ankara, TURKEY 3 Department of Mathematics Faculty of Arts and Sciences Çanakkale Onsekiz Mart University Çanakkale, TURKEY Abstract: In this paper, we investigate such a regular elements α and idempotents of the complete semigroup of binary relations B X D defined by semilattices of the class lower incomplete nets, for which VD,α = Q. Also we investigate right units of the semigroup B X Q. For the case where X is a finite set we derive formulas by means of which we can calculate the numbers of regular elements, idempotents and right units of the respective semigroup. AMS Subject Classification: 20M30, 20M10, 20M15 Key Words: semigroups, binary relation, regular element, idempotents, right units Received: January 30, 2014 Correspondence author c 2014 Academic Publications, Ltd. url:

2 550 Y. Diasamidze, A. Erdogan, N. Aydın 1. Introduction Let X be an arbitrary nonempty set. Let D be some nonempty set of subsets of the set X, closed with respect to the operation of set-theoretic union of elements of the set D, i.e., D D for any nonempty subset D of the set D. In that case, the set D is called complete X semilettice of unions. The union of all elements of the set D is denoted by the symbol D. Clearly, D D is the largest element. Recall thatabinaryrelation onthesetx isasubsetofthecartesianproduct X X. Ifαandβ arebinaryrelationsonthesetx withtheelements x,y,z X the condition x,y α is denoted as xαy and xαyβz means the conditions xαy and yβz are satisfied simultaneously. The binary relation α 1 = {x,y : yαx} is usually called the binary relation inverse to α. The empty binary relation which is empty subset of X X is denoted by. The binary relation δ = α β is called product of binary relations α and β. A pair x,y belongs to δ if only if there exists y X such that xαyβz. The binary operation is associative. So, B X, the set of all binaryrelations on X, is therefore asemigroup with respect to the operation. This semigroup is called the semigroup of all binary relations on the set X. Let f be an arbitrary mapping from X into D. Then one can construct such a mapping f with a binary relation α f on X provided by the condition below, α f = {x} fx. The set of all such binary relation is denoted x X by B X D. It is easy to prove that B X D is a semigroup with respect to the product operation of binary relations. This semigroup, B X D, is called a complete semigroup of binary relations defined by an X semilattice of unions D. Further, let x,y X, Y X, α B X D, T D, D D and t D. Then we have the following notation, yα = {x X : yαx},yα = yα,2 X = {Y : Y X}, X = 2 X \{ } y Y VD,α = {Yα : Y D}, D t = { Z : t Z }, D T = { Z D : T Z }, D T = { Z D : Z T }. Now, let s take α B X D. If β α = β for any β B X D, then α is called a right unit of semigroup B X D. If α α = α then α is called an idempotent element of semigroup B X D. And if α β α = α for some β B X D, then a binary relation α is called a regular element of semigroup B X D.

3 SOME REGULAR ELEMENTS, IDEMPOTENTS AND RIGHT D is partially ordered with respect to the set theoretic inclusion. Let D D and ND,D = {Z D : Z Z for any Z D }. It is clear that ND,D is the set of lower bounds of a nonempty subset D included in D. If ND,D then ND,D belongs to D and it is the greatest lower bound of D and is denoted by D,D = ND,D. Let ld,t = D \D T. We say that a nonempty element T is a nonlimiting element of D if T\lD,T. A nonempty element T is a limiting element of D if T\lD,T =. Now, we continue with some essential definitions and theorems given by the cited references. Definition 1.1. [2, Definition 1] Let α B X, T V X,α and YT α = {y X : yα = T}. Then a representation of a binary relation α of the form α = T is called quasinormal. T VX,α Y α T Note that, if α = T VX,α Y α T binary relation α then the following are true, 1. X = T VX,α Y α T 2. Y α T Yα T = for T,T VX,α and T T. T is a quasinormal representation of the Definition 1.2. [3, Definition 2] Let D and D be some nonempty subsets of the complete X semilattices of unions. We say that a subset D generates a set D if any element from D is a set-theoretic union of the elements from D. Definition 1.3. [4, Definition ] We say that a complete X semilattice of unions D is an XI semilattice of unions if it satisfies the following two conditions: a D,D t D for any t D, b Z = t Z D,D t for any nonempty element Z of D. Theorem 1.1. [4, Corollary ] Let Y = {y 1,y 2,...,y k } and D j = {T 1,...,T j } be some sets, where k 1 and j 1. Then the numbers sk,j of all possible mappings of the sets Y on any subset D j of the set D j and T j D j can be calculated by the formula sk,j = j k j 1 k.

4 552 Y. Diasamidze, A. Erdogan, N. Aydın Lemma 1.1. [1, Lemma 3.1] Let D complete X semilattices of unions. If a binary relation ε having the form ε = εd,f = {t} D,D t X\ D D t D is a right unit of the semigroup B x D, then it is the largest right unit of this semigroup. Theorem 1.2. [1, Theorem2.5] Let D beacompletesubsemilattice ofthe complete X semilattice of unions D, D = D and f be an arbitrary mapping of the set X\ D in the semilattice D. If D is a complete XI semilattice of unions then the binary relation α = αd,f = {t} D,D t { t } ft t D t X\ D is an idempotent element of semigroup B X D and VD,α = D. Theorem 1.3. [4, Theorem 4.1.3] A binary relation ε B X D is right units of this semigroup iff ε is idempotent and VD,ε = D. Definition 1.4. [6, Definition 7] A one-to-one mapping ϕ between the complete X semilattices of unions D and D is called a complete isomorphism if the condition ϕ D 1 = T D 1 ϕt is fulfilled for each nonempty subset D 1 of the semilattice D. Definition 1.5. [6, Definition8] Let α besomebinaryrelation ofthesemigroup B X D. We say that a complete isomorphism ϕ between XI semilattice of unions Q and D is a complete α isomorphism if a Q = VD,α b ϕ = for VD,α and ϕtα = T for any T VD,α. Theorem 1.4. [4, Theorem 6.3.3] Let D be a finite X semilattice of unions and α σ α = α for some elements α,σ B X D; Dα the set those elements T ofthesemilattice D = VD,α\{ }whicharenonlimitingelements of the set Dα T. Then binary relation α having a quasinormal representation of the form α = YT α T is a reguler element of the semigroup B XD T VD,α iff the set VD,α is a XI semilattice of unions and for some α isomorphism ϕ of the semilattice VD,α on some X subsemilattice D of the semilattice D the following conditions are fulfilled:

5 SOME REGULAR ELEMENTS, IDEMPOTENTS AND RIGHT i T Dα T Y α T ϕt for any T Dα; ii Y α T ϕt for any nonlimiting element T of the set Dα T. Theorem 1.5. [4, Theorem 6.3.7] A regular element α of the semigroup B X D is idempotent iff the mapping ϕ satisfying the condition ϕt = Tα for any T VD,α is an identity mapping of the semilattice VD,α. Definition 1.6. [4, Definition 6.3.4] Let Q and D be respectively some XI and X subsemilattices of the complete X semilattice of unions D. Then R ϕ Q,D is a subset of the semigroup B X D such that α R ϕ Q,D only if the following conditions are fulfilled for the elements of α and ϕ, a The binary relation α be regular element of the semigroup B X D, b VD,α = Q, c ϕ is a α isomorphism between the complete semilattices of unions Q and D satisfying the conditions i and ii of the Theorem 1.4. Definition 1.7. [4, Definition6.3.4] LetΦD,D bethesetofallcomplete isomorphism ϕ between XI semilattice of unions D and D such that ϕ ΦD,D only if ϕ is a α isomorphism for some α B X D and VD,α = D. ΩD is the set of all XI subsemilattices of the complete X semilat-tice of unions D such that D ΩD iff there exists a complete isomorphism between the semilattices D and D. Let us denote RD,D = ϕ ΦD,D R ϕ D,D and RD = D ΩD RD,D. Theorem 1.6. [4, Theorem 6.3.5] Let X is a finite set. If ϕ is a fixed element of the set ΦD,D and ΩD = m 0 and q is a numberof all automorhisms of the semilattice D, then RD = m 0 q R ϕ D,D. } Theorem 1.7. [6, Theorem 22] Let D = { D,T1,T 2,...,T m 1 be some finitex semilattice ofunionsandcd = {P 0,P 1,P 2,...,P m 1 }bethefamily of sets of pairwise disjoint subsets of the set X. If ϕ is a mapping of the semilattice D to the family sets CD that satisfies the condition ϕ D = P 0

6 554 Y. Diasamidze, A. Erdogan, N. Aydın and ϕt i = P i i = 1,2,...,m 1 and ˆD Z = D\{T D : Z T} then the following equalities are valid: D = P 0 P 1 P m 1, T i = P 0 T ˆD Ti ϕt. 1.1 In the sequel these equalities will be called formal. It is proved that if the elements of the semilattice D are represented in the form 1.1, then among the parameters P i i = 0,1,2,...,m 1 there exist such parameters that cannot be empty sets for D. Such sets P i 0 < i m 1 are called basis sources, whereas sets P j 0 j m 1 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. Note that the set P 0 is always considered to be a source of completeness. Lemma 1.2. Let D and CD = {P 0,P 1,...,P n 1 } are the finite semilattice of unions and the family of sets of pairwise nonintersecting subsets of the set X; ϕ is a mapping of the semilattice D on the family of sets CD. If ϕt = P CD\{P 0 } for some T D, then D t = D\ D T for all t P. Proof. Let t and Z are any elements of the set P P P 0 and of the semilattice D respectively. Then the equality P Z = i.e., Z / D t for any t P is true if and only if T / ˆD Z if T ˆD Z, then ϕt Z by definition of the formal equalities of the semilattice D. Since ˆD Z = D\{T D : Z T }by definition of theset ˆD Z. Thus thecondition T / ˆD Z hold iff T {T D : Z T }. So, Z T and Z D T by definition of the set D T. Therefore, ϕt Z = if and only if Z D T. Of this follows that the inclusion ϕt = P Z is true iff D t = D\ D T for all t ϕt = P.

7 SOME REGULAR ELEMENTS, IDEMPOTENTS AND RIGHT Results Let N m = {0,1,2,...,m} m 1 be some subset of the set of all natural numbers. A subsemilattice Q = {T ij X : i N s, j N k }\{T 00 } of the complete X semilattice of unions D is called lower incomplete net which contains two subsets Q 1 = {T 10,T 20,...T s0 }, Q 2 = {T 01,T 02,...,T 0k } and satisfies the following conditions: a T 10 T 20 T s0 and T 01 T 02 T 0k ; b Q 1 Q 2 = ; c T pq T ij if p,q i,j d the elements of the sets Q 1 and Q 2 are pairwise noncomparable; e T ij T i j = T pq, if p = max{i,i } and q = max{j,j }. Note that the diagram of the given X semilattice of unions Q is shown in Fig Let CQ = {P 01,P 10,P 02,...,P s 1k,P ks 1,P sk } is a family sets, where P 01, P 10, P 02,..., P ks 1, P sk are pairwise disjoint subsets of the set X and T01 T ϕ = 10 T 02 T 11 T T s 1k T ks 1 T sk P 01 P 10 P 02 P 11 P P s 1k P ks 1 P sk is a mapping of the semilattice Q onto the family sets CQ. Then for the formal equalities of Q we have a form see Theorem 1.7: T sk = P sk ϕt rt, 2.1 T rt ˆD sk T s 1k = P sk ϕt rt,t sk 1 = P sk ϕt rt, T rt ˆD s 1k T rt ˆD sk 1 T 20 = P sk T rt ˆD 20 ϕt rt, T 11 = P sk T rt ˆD 11 ϕt rt,t 02 = P sk T rt ˆD 02 ϕt rt,

8 556 Y. Diasamidze, A. Erdogan, N. Aydın T 10 = P sk T rt ˆD 10 ϕt rt = T 01 T 02 T 0k T 01 = P sk T rt ˆD 01 ϕt rt = T 10 T 20 T s0. Here P 0k, P 1k,..., P s 1k, P s0, P s1,..., P sk 1 are basis sources, the elements of the set CQ\{P 0k, P 1k, P 2k,..., P s0, P s1,..., P sk 1 } are sources of completeness of the semilattice Q see Theorem 1.7. Lemma 2.1. Let Q be a lower incomplete net. Then Q is XI semilattice of unions iff it satisfies the condition T 0k T s0 =. Proof. Let t Q, Q t = {Z Q : t Z} and Q,Q t is the exact lower bound of the set Q t in Q. Then fromlemma 1.2 and from the formal equalities 2.1 we have: t P sk, Q t = Q t P s 1k, Q t = Q\ Q Ts 1k, t P s 2k, Q t = Q\ Q Ts 2k, t P sk 1, Q t = Q\ Q Tsk 1, t P sk 2, Q t = Q\ Q Tsk 2, t P 0k, Q t = Q\ Q T0k, t P s0, Q t = Q\ Q Ts0, Q,Q t = T s0, if t P s 1k, T s 10, if t P s 2k T 0k, if t P sk 1, T 0k 1, if t P sk 2, T 10, if t P 0k, T 01, if t P s0. IfT ij T s 1k 1, ϕt ij = P ij andt P ij, thenq t = Q\ Q Tij and Q,Q t / Q. We have Qˆ = { Q,Q t : Q,Q t Q} = {T 10,...,T s0, T 01,...,T 0k } and Q,Q t / Q, when t / {P 0k,P 1k,...,P s 1k, P s0,p s1,...,p sk 1 }, i.e., Q,Q t / Q, when t T 0k T s0. Therefore the semilattice Q is not XIsemilattice of unions, if T 0k T s0. If T 0k T s0 =, i.e., t {P 0k,P 1k,...,P s 1k, P s0,... P sk 1 } then Q,Q t Q for all elements t of the set D and T ij = t T ij Q,Q t. Therefore the semilattice Q is XI-semilattice of unions, if T 0k T s0 =. If the equality T 0k T s0 = is true, then by follows that Q is XIsemilattice of unions. Lemma 2.2. Let Q be a XI lower incomplete net. Then following equalities are true: P s 1k = T s0 \T s 1k, P s 2k = T s 10 \T s 2k,..., P 0k = T 10 \T 0k

9 SOME REGULAR ELEMENTS, IDEMPOTENTS AND RIGHT P sk 1 = T 0k \T sk 1, P sk 2 = T 0k 1 \T sk 2,..., P s0 = T 01 \T s0. Proof. The given Lemma immediately follows from the formal equalities 2.1 of the semilattice Q. For the largest right unit ε of the semigroup B X D we have: ε = P s 1k T s0 P s 2k T s 10 P sk 1 T 0k T 0k 1 \Y sk 2 T 0k 1 P 0k T 10 P s0 T 01 = T s0 \T s 1k T s0 T s 10 \T s 2k T s 10 T 0k \T sk 1 T 0k T 0k 1 \T sk 2 T 0k 1 T 10 \T 0k T 10 T 01 \T s0 T 01 see, Lemma 1.1. Theorem 2.1. Let Q be a XI-lower incomplete net. Then a binary relation α of the semigroup B X D having a quasinormal representation of the form α = Yij α T ij such that Q = VD,α, is a regular element of the T ij Q semigroup B X D iff for some α isomorphism ϕ of the semilattice Q on some subsemilattice D of the semilattice D the following conditions are fulfilled: Y00 α ϕt 0k ϕt s0, Y00 α Y01 α ϕt 01, Y00 α Y01 α Y02 α ϕt 02,..., Y00 α Yα 01 Yα 02 Yα 0k ϕt 0k, Y00 α Yα 10 ϕt 10,Y00 α Yα 10 Yα 20 ϕt 20,..., Y00 α Y10 α Y20 α Ys0 α ϕt s0, Yij α ϕt ij for any T ij Q 1 Q 2 \{ }. Proof. It is easy to see that the set Qˆ = {T 10,T 20,...,T s0,t 01,...,T 0k } is an irreducible generating set of the semilattice Q. Moreover, all elements of the set Qˆ = Q 1 Q 2 are nonlimiting. Now by Theorem 1.4 we obtain a ϕt pq for any T pq Qˆ. T ij QˆTpq b Yij α ϕt ij for any T ij Qˆ. From the condition a of this theorem we immediately have the validity of the follo-wing inclusions: Y10 α ϕt 10, Y10 α Yα 20 ϕt 20,..., Y10 α Yα 20 Yα s0 ϕt s0 Y01 α ϕt 01, Y01 α Yα 02 ϕt 02,..., Y01 α Yα 02 Yα 0k ϕt 0k Y20 α ϕt 20,..., Ys0 α ϕt s0, Y02 α ϕt 02,...,Y0k α ϕt 0k,

10 558 Y. Diasamidze, A. Erdogan, N. Aydın Theorem 2.2. Let Q be a XI lower incomplete net. Then a binary relation α of the semigroup B X Q, which has a quasinormal representation of the form α = Yij α T ij such that Q = VD,α, is an idempotent T ij Q element of the semigroup B X D iff the following conditions are fulfilled: Y10 α T 10, Y10 α Y20 α T 20,..., Y10 α Y20 α Ys0 α T s0, Y01 α T 01, Y01 α Y02 α T 02,..., Y01 α Y02 α Y0k α T 0k, Y20 α T 20,..., Ys0 α T s0, Y02 α T 02,...,Y0k α T 0k, for any T ij Qˆ. Proof. This theorem immediately follows from Lemma 2.1, from the Theorem 2.1 and Theorem 1.5. Theorem 2.3. Let Q be a XI lower incomplete net. Then a binary relation α of the semigroup B X Q that has a qusinormal representation of the form α = Yij α T ij such that Q = VD,α, is a right unit of the T ij Q semigroup B X Q iff the following conditions are fulfilled: Y10 α T 10, Y10 α Y20 α T 20,..., Y10 α Y20 α Ys0 α T s0, Y01 α T 01, Y01 α Yα 02 T 02,..., Y01 α Yα 02 Yα 0k T 0k, Y20 α T 20,..., Ys0 α T s0, Y02 α T 02,...,Y0k α T 0k, for any T ij Qˆ. Proof. This theorem immediately follows from Theorem 1.3. Theorem 2.4. Let Q be a XI lower incomplete net. If the semilattice Q and D = { T01, T 10,..., T } sk see Fig. 2.2 are α isomorphic and ΩQ = m0, then the following equalities are valid: a RD = m0 j T 02\T s1 1 3 T 03\T s2 2 T 03\T s2 k 1 T 0k 1\T sk 2 k 2 T 0k 1\T sk 2 k T 0k\T sk 1 k 1 T 0k\T sk 1 2 T 20\T 1k 1 3 T 30\T 2k 2 T 30\T 2k... s 1 T s 10\T s 2k i 1 T s 10\T s 2k

11 SOME REGULAR ELEMENTS, IDEMPOTENTS AND RIGHT Q s T s0\t s 1k s 1 T s0\t s 1k X\ T sk, if s k or b RD = 2 m0 j T 02\T s1 1 3 T 03\T s2 2 T 03\T s2 k 1 T 0k 1\T sk 2 k 2 T 0k 1\T sk 2 k T 0k\T sk 1 k 1 T 0k\T sk 1 2 T 20\T 1k 1 3 T 30\T 2k 2 T 30\T 2k... s 1 T s 10\T s 2k i 1 T s 10\T s 2k Q s T s0\t s 1k s 1 T s0\t s 1k X\ T sk, if s = k. Proof. In the first place, we note that the given semilattice Q has one automorphisms i.e., ΦQ,Q = 1 if s k and two automorphism i.e., ΦQ,Q = 2 if s = k see [4, Theorem ]. Hence, we have ΦQ,D = 1 or ΦQ,D = 2 see [4, Lemma 6.3.2]. Next, assume that α RQ,D and a quasinormal representation of a regular binary relation α has the form α = T ij Q Then, according to Theorem 2.1, we have Y α ij T ij 2.2 Y10 α T 10, Y10 α Y20 α T 20,..., Y10 α Y20 α Ys0 α T s0, 2.3 Y01 α T 01, Y01 α Y02 α T 02,..., Y01 α Y02 α Y0k α T 0k, Y20 α T 20,..., Ys0 α T s0, Y02 α T 02,...,Y0k α T 0k, for any T ij { T01, T 02,..., T 0k, T 10, T 20,..., T s0 }. Further, let f α be a mapping the set X in the semilattice D satisfying the conditions f α t = tα for all t X. f 01α,f 02α,...,f 0k 1α,f 0kα, f 10α,f 20α,...,f s 10α,f s0α and f skα, are the restrictions of the mapping f α, on the sets T 01, T 02 \ T s1,..., T 0k 1 \ T sk 2, T0k \ T sk 1,

12 560 Y. Diasamidze, A. Erdogan, N. Aydın T 10, T20 \ T 1k,..., Ts 10 \ T s 2k, Ts0 \ T s 1k, X\ T sk respectively. It is clear that the intersection disjoint elements of the set { are empty set, and T 01, T 02 \ T s1,..., T } 0k 1 \ T sk 2, T0k \ T sk 1, T 10, T20 \ T 1k,..., Ts 10 \ T s 2k, Ts0 \ T s 1k, X\ T sk T 01 T02 \ T s1 T0k 1 \ T sk 2 T0k \ T sk 1 T 10 T20 \ T 1k Ts 10 \ T s 2k Ts0 \ T s 1k X\ Tsk = X. We are going to find properties of the maps f 01α, f 02α,..., f 0k 1α, f 0kα, f 10α, f 20α,..., f s 10α, f s0α and f skα,. 1 t T 01. Then by properties 2.3 we have Y01 α T 01, i.e., t Y01 α and tα = T 01 by definition of the set Y01 α. Therefore f 01αt = T 01 for all t T t T 0j \ T sj 1 j = 2,...,k 1,k. Then by properties 2.3 we have T 0j \ T sj 1 T 0j Y01 α Y 01 α Y 0j α, i.e., t Y 01 α Y 01 α Y 0j α and tα {T 01,T 02,...,T 0j } by definition of the sets Y01 α,y 01 α,...,y 0j α. Therefore f 0jα t {T 01,T 02,...,T 0j } for all t T 0j \ T sj 1. By suppose we have that Y0j α T 0j, i.e., t 0j α = T 0j for some t 0j T 0j. If t oj T sj 1, then T sj 1 = T s0 T 0j 1 by definition of the non complete net, and t 0j T sj 1 = T s0 T 0j 1 Y α 10 Y α 20 Y α s0 Y α 01 Y α 0j 1.

13 SOME REGULAR ELEMENTS, IDEMPOTENTS AND RIGHT So, t 0j α {T 10,T 20,...,T s0,t 01,T 02,...,T 0j 1 } by definition of the sets Y10 α, Y 20 α,..., Y s0 α, Y 01 α, Y 02 α,..., Y 0j 1 α. The condition t 0j α {T 10, T 20,..., T s0, T 01, T 02,..., T 0j 1 } contradict of the equality t 0j α T 0j, whilet 0j / {T 10,T 20,...,T s0, T 01,...,T 0j 1 }. Therefore,f 0jα t 0j = T 0j for some t 0j T 0j \ T sj 1. 3 t T 10. Then by properties 2.3 we have Y10 α T 10, i.e., t Y10 α and tα = T 10 by definition of the set Y10 α. Therefore f 10αt = T 10 for all t T t T i0 \ T i 1k j = 2,...,s 1,s. Then by properties 2.3 we have T i0 \ T i 10 T i0 Y10 α Y 20 α Y i0 α, i.e., t Y 10 α Y 20 α Y i0 α and tα {T 10,T 20,...,T i0 } by definition of the sets Y10 α,y 20 α,...,y i0 α. Therefore f i0α t {T 10,T 20,...,T i0 } for all t T i0 \ T i 1k. By suppose we have that Yi0 α T i0, i.e., t i0α α = T i0 for some t i0 T i0. If t i0 T i 1k, then T i 1k = T i 10 T 0k by definition of the non complete net, and t i0 T i 1k = T i 10 T 0k Y10 α Yα 20 Yα i 10 Y α 01 Y0k α. So, t i0 α {T 10,T 20,...,T i 10,T 01,T 02,...,T 0k } by definition of the sets Y10 α, Y 20 α,..., Y i 10 α, Y 01 α, Y 02 α,..., Y 0k α. The condition t i0α α {T 10, T 20,..., T i 10, T 01, T 02,..., T 0k } contradict of the equality t i0α α = T i0, while T i0 / {T 10,T 20,...,T i 10,T 01,..., T 0k }. Therefore, f i0α t i0α = T i0 for some t i0 T i0 \ T i 1k. 5 t X\ T sk. Thenby definitionquasinormalrepresentation binaryrelation αandbyproperty2.3wehavet X\ T sk X = Y ij i,j / {0,0}, i N s,j N k i.e., tα QbydefinitionofthesetsYij α. Thereforef skαt Qforall t X\ T sk. Therefore, for every binary relation α RQ,D there exists an ordered system Further, let f 01α,f 02α,...,f 0k 1α,f 0kα,f 10α,f 20α,...,f s 10α,f s0α,f skα. 2.4 f 01 : T 01 {T 01 }, f 10 : T 10 {T 10 }, f 0j : T 0j \ T sj 1 {T 00,T 01,T 02,...,T 0j }, j = 2,...,k 1,k, f i0 : T i0 \ T i 1k {T 00,T 10,T 20,...,T i0 }, i = 2,...,s 1,s, f sk : X\ T sk {T ij : i N s,j N k },

14 562 Y. Diasamidze, A. Erdogan, N. Aydın are such mappings, which satisfying the conditions: 6 f 01 t = T 01 for all t T 01 ; 7 f 10 t = T 10 for all t T 10 ; 8 f 0j t {T 01,T 02,...,T 0j } for all t T 0j \ T sj 1 and f 0j t 0j = T 0j for some t 0j T 0j \ T sj 1 ; 9 f i0 t {T 10,T 20,...,T i0 } for all t T i0 \ T i 1k and f i0 t i0 = T i0 for some t i0 T i0 \ T i 1k ; 10 f sk t Q for all t X\ T sk. Now, we define a map f of a set X in the semilattice D, which satisfies the condition: f 01 t, if t T 01 f oj t, if t T 0j \ T sj 1 j = 2,...,k 1,k ft = f 10 t, if t T 10 f i0 t, if t T i0 \ T i 1k i = 2,...,s 1,s f sk t, if t X\ T sk. Further, let β = {x} f x, and Y β ij = {t : tβ = T ij} T ij Q. Then x X binary relation β may be representation by form β = Y β ij T ij and satisfying the conditions: T ij Q Y10 α T 10, Y10 α Yα 20 T 20,..., Y10 α Yα 20 Yα s0 T s0, Y01 α T 01, Y01 α Yα 02 T 02,..., Y01 α Yα 02 Yα 0k T 0k, Y20 α T 20,..., Ys0 α T s0, Y02 α T 02,...,Y0k α T 0k, By suppose f i0 t i0 = T i0 for some t i0 T i0 \ T i 1k and f 0j t 0j = T 0j for some t 0j T 0j \ T sj 1. From this and by Theorem 2.1 we have that β RQ,D. Therefore for every binary relation α RQ,D and ordered system 2.4 exist one to one mapping. By the Theorem 1.1 the number of the mappings f 01α, f 02α,..., f 0k 1α, f 0kα, f 10α, f 20α,..., f s 10α, f s0α, f skα are respectively: 1,j T 02\T s1 1,3 T 03\T s2 2 T 03\T s2,...,k 1 T 0k 1\T sk 2 k 2 T 0k 1\T sk 2, k T 0k\T sk 1 k 1 T 0k\T sk 1,2 T 20\T 1k 1, 3 T 30\T 2k 2 T 30\T 2k,..., s 1 T s 10\T s 2k i 1 T s 10\T s 2k,s T s0\t s 1k s 1 T s0\t s 1k, D X\ T sk. Therefore the equality RQ,D = 2 T 02\T s1 1 3 T 03\T s2 2 T 03\T s2

15 SOME REGULAR ELEMENTS, IDEMPOTENTS AND RIGHT k 1 T 0k 1\T sk 2 k 2 T 0k 1\T sk 2 k T 0k\T sk 1 k 1 T 0k\T sk 1 3 T 30\T 2k 2 T 30\T 2k... s 1 T s 10\T s 2k i 1 T s 10\T s 2k Q s T s0\t s 1k s 1 T s0\t s 1k X\ T sk, 2 T 20\T 1k 1 is valid. Now, using the equalities ΩQ = m 0 we Obtain RD = m0 2 T 02\T s1 1 3 T 03\T s2 2 T 03\T s2 k 1 T 0k 1\T sk 2 k 2 T 0k 1\T sk 2 k T 0k\T sk 1 k 1 T 0k\T sk 1 2 T 20\T 1k 1 3 T 30\T 2k 2 T 30\T 2k... s 1 T s 10\T s 2k i 1 T s 10\T s 2k Q s T s0\t s 1k s 1 T s0\t s 1k X\ T sk, if s k or 2 T 02\T s1 1 3 T 03\T s2 2 T 03\T s2 RD = 2 m0 k 1 T 0k 1\T sk 2 k 2 T 0k 1\T sk 2 k T 0k\T sk 1 k 1 T 0k\T sk 1 3 T 30\T 2k 2 T 30\T 2k... Q s T s0\t s 1k s 1 T s0\t s 1k X\ T sk, if s = k see Theorem T 20\T 1k 1 s 1 T s 10\T s 2k i 1 T s 10\T s 2k Corollary 2.1. Let Q be a XI lower incomplete net and E r X Q be the set of all right units of the semigroup B x Q. If X is a finite set, then the following formula is true E r X Q = 2 T 02\T s1 1 3 T 03\T s2 2 T 03\T s2

16 564 Y. Diasamidze, A. Erdogan, N. Aydın k 1 T 0k 1\T sk 2 k 2 T 0k 1\T sk 2 k T 0k\T sk 1 k 1 T 0k\T sk 1 2 T 20\T 1k 1 3 T 30\T 2k 2 T 30\T 2k... s 1 T s 10\T s 2k i 1 T s 10\T s 2k Q s T s0\t s 1k s 1 T s0\t s 1k X\ T sk. Proof. By virtue of Theorem [4, Theorem ] see [4] we have E r X Q = R ε Q Q,Q, where ε Q is the identity mapping of the net Q. Now, taking into account Theorem 1.5 and Theorem 2.4, we obtain the validity of corollary. Corollary 2.2. Let Q = {T 01,T 10,T 11 } be a XI subsemilattice of the semilattice D see Fig If the semilattices Q and D = { T01, T 10, T 11 } are α isomorphic, ΩQ = m 0, then the following equality is valid: RD = 2 m 0 3 X\ T 11. Proof. It is obvious that in this case ΦQ,D = 2. Thereforethe corollary immediately follows from Theorem 2.4. Corollary 2.3. Let Q = {T 01,T 10,T 02,T 11,T 20,T 12,T 21,T 22 } be a XI subsemilattice of the semilattice D see Fig If the semilatices Q and D = { T01, T 10, T 02, T 11, T 20, T 12, T 21, T 22 } are α isomorphic, ΩQ = m 0, then the following equality is valid: RD = 2 m0 2 T 20 \ T T 02 \ T X\ T 22. Proof. It is obvious that in this case ΦQ,D = 2. Thereforethe corollary immediately follows from Theorem 2.4.

17 SOME REGULAR ELEMENTS, IDEMPOTENTS AND RIGHT Corollary 2.4. Let Q = {T 01,T 10,T 02,T 11,T 20,...,T 33 } be a XI subsemilattice of the semilattice D see Fig If the semilatices Q and D = { T01, T 10, T 02, T 11, T 20,..., T 33 } are α isomorphic, ΩQ = m 0, then the following equality is valid: RD = 2 m0 2 T 20 \ T T 30 \ T 23 2 T 30 \ T T 02 \ T T 03 \ T 32 2 T 03 \ T 32 X\ T 33. Proof. It is obvious that in this case ΦQ,D = 2. Thereforethe corollary immediately follows from Theorem 2.4. Corollary 2.5. Let Q = {T 01,T 10,T 02,T 11,T 20,...,T s 11,T s0,t s1 } be a XI sub-semilattice of the semilattice D see Fig If the semilattices Q and D = { T01, T 10, T 02, T 11, T 20,..., T s 11, T s0, T s1 } are α isomorphic, ΩQ = m 0, then the following equality is valid: RD = m0 2 T 20 \ T T 30 \ T 21 2 T 30 \ T 21 s 1 T s 10 \ T s 21 s 2 T s 10 \ T s 21 2 s+1 1 s T s0 \ T s 11 s 1 T s0 \ T s 11 X\ T s1. Proof. It is obvious that in this case ΦQ,D = 1. Thereforethe corollary immediately follows from Theorem 2.4. Corollary 2.6. Let Q = {T 01,T 10,T 11,T 20,T 21 } be a XI subsemilattice of the semilattice D see Fig If the semilattices Q and D = { T01, T 10, T 11, T 20, T 21 }

18 566 Y. Diasamidze, A. Erdogan, N. Aydın are α isomorphic, ΩQ = m 0, then the following equality is valid: RD = m 0 2 T 20 \ T X\ T 21. Proof. It is obvious that in this case ΦQ,D = 1. Thereforethe corollary immediately follows from Theorem 2.4. Corollary 2.7. Let Q = {T 01,T 10,T 11,T 20,T 21,T 30,T 31 } be a XI subsemilattice of the semilattice D see Fig If the semilattices Q and D = { T01, T 10, T 11, T 20, T 21, T 30, T 31 } are α isomorphic, ΩQ = m 0, then the following equality is valid: RD = m T 20 \ T T 30 \ T 21 2 T 30 \ T 21 X\ T 31. Proof. It is obvious that in this case ΦQ,D = 1. Thereforethe corollary immediately follows from Theorem 2.4. References [1] Ya. Diasamidze, Complete Semigroups of Binary Relations. Journal of Mathematical Sciences, Plenum Publ. Cor., New York, Vol. 117, No. 4, 2003, [2] Ya. Diasamidze, Sh. Makharadze and G. Fartenadze, Maximal Subgroups of Complete Semigroups of Binary Relations. Proceedigs of A. Razmadze Mathematical Institute, Vol. 131, 2003, [3] Ya. Diasamidze, To the The Theory of the Binary Relation Semigroups. Proceedings of A. Razmadze Mathematical Institute, Vol. 128, 2002, [4] Ya. Diasamidze and Sh. Makharadze, Complete Semigroups of Binary Relations, Kriter Yayınevi, İstanbul [5] Ya. Diasamidze, Sh. Makharadze, G. Partenadze and O. Givradze, On finite X - semilattices of unions. Journal of Mathematical Sciences, Plenum Publ. Cor., New York, Vol. 141, No. 4, 2007, [6] Ya. Diasamidze and Sh. Makharadze, Complete Semigroups of Binary Relations Defined by X- Semilattices of Unions. Journal of Mathematical Sciences, Plenum Publ. Cor., New York, Vol. 166, No. 5, 2010,