International Journal of Pure and Applied Mathematics Volume 86 No. 1 2013, 199-216 ISSN: 1311-8080 (printed version); ISSN: 1314-3395 (on-line version) url: http://www.ijpam.eu AP doi: http://dx.doi.org/10.12732/ijpam.v86i1.13 ijpam.eu REGULAR ELEMENTS OF THE COMPLETE SEMIGROUPS OF BINARY RELATIONS OF THE CLASS 7(X, 8) P Barı¸sAlbayrak1, I. Yasha Diasamidze2, Ne¸set Aydin3 1,3Department of Mathematics Faculty of Science and Art Canakkale Onsekiz Mart University Canakkale, TURKEY 2Shota Rustaveli State University 35, Ninoshvili St., Batumi 6010, GEORGIA Abstract: In this paper let Q = {T1,T2,T3,T4,T5,T6,T7,T8} be a subsemi- lattice of X−semilattice of unions D where T1 ⊂ T2 ⊂ T3 ⊂ T5 ⊂ T6 ⊂ T8, T1 ⊂ T2 ⊂ T3 ⊂ T5 ⊂ T7 ⊂ T8,T1 ⊂ T2 ⊂ T4 ⊂ T5 ⊂ T6 ⊂ T8,T1 ⊂ T2 ⊂ T4 ⊂ T5 ⊂ T7 ⊂ T8,T1 6= ∅,T4\T3 6= ∅,T3\T4 6= ∅,T6\T7 6= ∅,T7\T6 6= ∅, T3 ∪ T4 = T5, T6 ∪ T7 = T8, then we characterize the class each element of which is isomorphic to Q by means of the characteristic family of sets, the character- istic mapping and the generate set of Q. Moreover, we calculate the number of regular elements of BX (D) for a finite set X. AMS Subject Classification: 20M30, 20M10, 20M15 Key Words: semigroups, binary relation, regular elements 1. Introduction Let X be an arbitrary nonempty set. Recall that a binary relation on X is a subset of the cartesian product X × X. The binary operation ◦ on BX (the set c 2013 Academic Publications, Ltd. Received: May 8, 2013 url: www.acadpubl.eu 200 B. Albayrak, I.Y. Diasamidze, N. Aydin of all binary relations on X) defined by for α, β ∈ BX (x, z) ∈ α ◦ β ⇔ (x, y) ∈ α and (y, z) ∈ β, for some y ∈ X is associative. Therefore BX is a semigroup with respect to the operation ◦. This semigroup is called the semigroup of all binary relations on the set X. Let D be a nonempty set of subsets of X which is closed under the union i.e., ∪D′ ∈ D for any nonempty subset D′ of D. In that case, D is called a complete X− semilattice of unions. The union of all elements of D is denoted by the symbol D.˘ Clearly, D˘ is the largest element of D. Let X be an arbitrary nonempty set and m be an arbitrary cardinal number. Σ(X, m) is the class of all complete X− semilattices of unions of power m. 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 e a set-theoretic union of the elements from D. e Note that the semilattice D is partially ordered with respect to the set- e theoretic inclusion. Let ∅ 6= D′ ⊆ D and N(D, D′) = Z ∈ D | Z ⊆ Z′ for any Z′ ∈ D′ . It is clear that N(D, D′) is the set of all lower bounds of D′. If N(D, D′) 6= ∅ then Λ(D, D′) = ∪N(D, D′) belongs to D and it is the greatest lower bound of D′. ′ Further, let x, y ∈ X, Y ⊆ X, α ∈ BX ,T ∈ D, ∅ 6= D ⊆ D and t ∈ D.˘ Then we have the following notations, yα = {x ∈ X | (y, x) ∈ α} , Y α = yα, y[∈Y ′ ′ V (D, α) = {Y α | Y ∈ D} ,Dt = {Z ∈ D | t ∈ Z } , .. ′ ′ ′ ′ ′ ′ ′ ′ DT = {Z ∈ D | T ⊆ Z } , DT = {Z ∈ D | Z ⊆ T } . Let f be an arbitrary mapping from X into D. Then one can construct a binary relation αf on X by αf = ({x} × f(x)) .The set of all such binary x[∈X relations is denoted by BX (D). It is easy to prove that BX (D) is a semigroup with respect to the operation ◦. In this case BX (D), is called a complete semigroup of binary relations defined by an X−semilattice of unions D. This structure was comprehensively investigated in Diasemidze [6]. If α ◦ β ◦ α = α for some β ∈ BX (D) then a binary relation α is called a regular element of BX (D). REGULAR ELEMENTS OF THE COMPLETE SEMIGROUPS... 201 α Let α ∈ BX ,YT = {y ∈ X | yα = T } and V (X∗, α), if ∅ ∈/ D, V [α] = V (X∗, α), if ∅ ∈ V (X∗, α), V (X∗, α) ∪ {∅} , if ∅ ∈/ V (X∗, α) and ∅ ∈ D. α Then a representation of a binary relation α of the form α = (YT × T ) T ∈[V [α] α is called quasinormal. Note that, if α = (YT × T ) is a quasinormal T ∈[V [α] α α α representation of the binary relation α, then X = YT and YT ∩YT ′ 6= ∅ T ∈V[(X∗,α) for T,T ′ ∈ V (X∗, α) which T 6= T ′. In [7] they show that, if β is regular element of BX (D), then V [β] = V (D, β) and a complete X−semilattice of unions D is an XI− semilattice of unions if Λ(D, Dt) ∈ D for any t ∈ D˘ and Z = Λ(D, Dt) for any nonempty element Z of D. t[∈Z Let D′ be an arbitrary nonempty subset of the complete X−semilattice of unions D. A nonempty element T ∈ D′ is a nonlimiting element of D′ if ′ ′ ′ ′ T \l(D ,T ) = T \ ∪ (D \DT ) 6= ∅. A nonempty element T ∈ D is limiting element of D′ if T \l(D′,T ) = ∅. The family C(D) of pairwise disjoint subsets of the set D˘ = ∪D is the characteristic family of sets of D if the following hold a) ∩D ∈ C(D) b) ∪C(D) = D˘ c) There exists a subset CZ(D) of the set C(D) such that Z = ∪CZ(D) for all Z ∈ D. A mapping θ : D → C(D) is called characteristic mapping if Z = (∩D) ∪ θ (Z′) for all Z ∈ D. Z[′∈Dˆ The existence and the uniqueness of characteristic family and characteristic mapping is given in Diasemidze [8]. Moreover, it is shown that every Z ∈ D can be written as Z = θ(Q˘)∪ θ (T ) , where Qˆ (Z) = Q\{T ∈ Q | Z ⊆ T }. T ∈[Qˆ(Z) A one-to-one mapping ϕ between two complete X− semilattices of unions ′ ′′ ′ D and D is called a complete isomorphism if ϕ(∪D1) = ′ ∪ ϕ(T ) for each T ∈D1 202 B. Albayrak, I.Y. Diasamidze, N. Aydin ′ nonempty subset D1 of the semilattice D . Also, let α ∈ BX (D). A complete isomorphism ϕ between XI−semilattice of unions Q and D is called a complete α− isomorphism if Q = V (D, α) and ϕ(∅) = ∅ for ∅ ∈ V (D, α) and ϕ(T )α = T for any T ∈ V (D, α). Let Q and D′ are respectively some XI and X− subsemilattices of the complete X− semilattice of unions D. Then ′ Rϕ Q, D = {α ∈ BX (D) | α regular element, ϕ complete α − isomorphism} where ϕ : Q → D′ complete isomorphism and V (D, α) = Q. Besides, let us denote ′ ′ ′ ′ ′ R(Q, D ) = Rϕ(Q, D ) and R(D ) = R(Q ,D ). ϕ∈Φ([Q,D′) Q′∈[Ω(Q) where ′ ′ Φ Q, D = ϕ | ϕ : Q → D is a complete α−isomorhism ∃α ∈ BX (D) , Ω(Q) = Q′ | Q′ is XI−subsemilattices of D which is complete isomorphic to Q . E. Schr¨oder described the theory of binary relations in detail in the 1890s ([1]). The basic concepts and the properties of the theory were introduced in ”Principia mathemetica” Whitehead and Russell([2]). The theory of binary re- lations has been improved by Riguet ([3] − [4]). Many researcher studied this theory using partial transformations as Vagner did ([5]) . Regular elements of semigroup play an importent role in semigroup theory. Therefore Diasamidze generate systmatic rules for understanding structure of a semigroup of binary re- lations and characterization of regular elements of these semigroup in ([6] − [9]). In general he studied semigroups but, in particular, he investigates complete semigroups of the binary relations. In this paper, we take in particular, Q = {T1,T2,T3,T4,T5,T6,T7,T8} sub- semilattice of X−semilattice of unions D where the elements Ti’ s, i = 1, 2,..., 8 are satisfying the following properties, T1 ⊂ T2 ⊂ T3 ⊂ T5 ⊂ T6 ⊂ T8, T1 ⊂ T2 ⊂ T3 ⊂ T5 ⊂ T7 ⊂ T8,T1 ⊂ T2 ⊂ T4 ⊂ T5 ⊂ T6 ⊂ T8,T1 ⊂ T2 ⊂ T4 ⊂ T5 ⊂ T7 ⊂ T8,T1 6= ∅,T4\T3 6= ∅,T3\T4 6= ∅,T6\T7 6= ∅,T7\T6 6= ∅, T3 ∪T4 = T5, T6 ∪T7 = T8. We will investigate the properties of regular elemant α ∈ BX (D) satisfying V (D, α) = Q. Moreover, we will calculate the number of regular elements of BX (D) for a finite set X. REGULAR ELEMENTS OF THE COMPLETE SEMIGROUPS... 203 As general, we study the properties and calculate the number of regular ′ ′ elements of BX (D) satisfying V (D, α) = Q where Q is a semilattice isomorph to Q. So, we characterize the class for each element of which is isomorphic to Q by means of the characteristic family of sets, the characteristic mapping and the generate set of D. 2. Preliminaries Theorem 2.1. [9, Theorem 10] Let α and σ be binary relations of the semigroup BX (D) such that α ◦ σ ◦ α = α. If D(α) is some generating set α of the semilattice V (D, α)\ {∅} and α = (YT × T ) is a quasinormal T ∈V[(D,α) representation of the relation α, then V (D, α) is a complete XI− semilattice of unions. Moreover, there exists a complete α−isomorphism ϕ between the semilattice V (D, α) and D′ = {T σ | T ∈ V (D, α)} , that satisfies the following conditions: a) ϕ (T ) = T σ and ϕ (T ) α = T for all T ∈ V (D, α) α b) YT ′ ⊇ ϕ(T ) for any T ∈ D(α), .