Let SLX be the free semilattice on a finite nonempty set X . There exists an undirected graph Γ(SLX ) associated with SLX whose vertices are the proper subsets of X , except the empty set, and two distinct vertices A and B of Γ(SLX ) are adjacent if and only if A ∪ B = X . In this paper, the diameter, radius, girth, degree of any vertex, domination number, independence number, clique number, chromatic number, and chromatic index of Γ(SLX ) have been established. Moreover, we have determined when Γ(SLX ) is a perfect graph and when the core of Γ(SLX ) is a Hamiltonian graph.

Let Tn and Sn be the full transformation semigroup and the symmetric group on Xn = {1,. .. , n} , respectively. For n, m ∈ Z + with m ≤ n − 1 let T (n,m) = {α ∈ Tn : Xmα = Xm}. In this paper we research generating sets and the rank of T (n,m). In particular, we prove that rank (T (n,m)) =    2 if (n, m) = (2, 1) or (3, 2) 3 if (n, m) = (3, 1) or 4 ≤ n and m = n − 1 4 if 4 ≤ n and 1 ≤ m ≤ n − 2. for 1 ≤ m ≤ n − 1 .

Let C n be the Catalan monoid on X n = {1,. .. , n} under its natural order. In this paper, we describe the sets of left zero-divisors, right zero-divisors and two sided zero-divisors of C n ; and their numbers. For n ≥ 4, we define an undirected graph Γ(C n) associated with C n whose vertices are the two sided zero-divisors of C n excluding the zero element θ of C n with distinct two vertices α and β joined by an edge in case αβ = θ = βα. Then we first prove that Γ(C n) is a connected graph, and then we find the diameter, radius, girth, domination number, clique number and chromatic numbers and the degrees of all vertices of Γ(C n). Moreover, we prove that Γ(C n) is a chordal graph, and so a perfect graph.

Let + denotes the set of all positive integers. Let = {1,2, … , } be the finite chain for ∈ + and let be the full transformation semigroup on. Also let and be the semigroup of order-preserving full contraction mappings, and the semigroup of order-preserving or order-reversing full contraction mappings on , respectively. It is well-known that and are subsemigroups of. In this paper we obtain ranks of the semigroups and .

Let 𝑛 ∈ ℤ + and 𝑋 𝑛 = {1,2, … , 𝑛} be a finite set. Let 𝐷 𝑛 be the order-decreasing full transformation semigroup on 𝑋 𝑛 . In this paper, we find the left zero-divisors, the right zero-divisors and two sided zerodivisors of 𝐷 𝑛 . Moreover, for 𝑛 ≥ 4 we define an undirected graph Γ(𝐷 𝑛 ) whose vertices are two-sided zero divisors of 𝐷 𝑛 excluding the zero element 𝜃 of 𝐷 𝑛 . In the graph, distinct two vertices 𝛼 and 𝛽 are adjacent if and only if 𝛼𝛽 = 𝜃 = 𝛽𝛼. In this paper, we prove that Γ(𝐷 𝑛 ) is a connected graph, and we find diameter, girth, the degrees of all vertices, the maximum degree and the minimum degree in Γ(𝐷 𝑛 ). Moreover, we give lower bounds for clique number and choromatic number of Γ(𝐷 𝑛 ). Sıra Azaltan Dönüşüm Yarıgruplarının Sıfır-Bölen ÇizgesiAnahtar kelimeler Sıfır-bölen çizge; Sıra azaltan dönüşümler; Çap; Klik sayısı Öz 𝑛 ∈ ℤ + olmak üzere 𝑋 𝑛 = {1,2, … , 𝑛} sonlu bir küme olsun. 𝑋 𝑛 üzerindeki tüm sıra azaltan dönüşümlerin yarıgrubu 𝐷 𝑛 olsun. Bu çalışmada 𝐷 𝑛 yarıgrubunun sol sıfır bölenleri, sağ sıfır bölenleri ve iki-yönlü sıfır bölenleri bulunmuştur. Ayrıca, 𝑛 ≥ 4 için köşeleri 𝐷 𝑛 yarıgrubunun sıfır elemanı 𝜃 dışındaki iki-yönlü sıfır bölenleri olmak üzere Γ(𝐷 𝑛 ) yönsüz çizgesi tanımlanmıştır. Bu çizgede 𝛼 ve 𝛽 farklı köşeler olmak üzere bu iki köşenin çizgede bir kenar oluşturması için gerek ve yeter koşul 𝛼𝛽 = 𝜃 = 𝛽𝛼 olmasıdır. Bu çalışmada Γ(𝐷 𝑛 ) çizgesinin bağlantılı olduğu ispatlanmış olup, çizgenin çapı, çizgedeki en kısa devir uzunluğu, tüm köşelerin dereceleri, en büyük derece ve en küçük derece bulunmuştur. Ayrıca, Γ(𝐷 𝑛 ) çizgesinde klik ve kromatik sayıları için bir alt sınır bulunmuştur.