The zero divisor graph of a special subsemigroup of order preserving tranformation semigroup

In this thesis, the zero-divisor graphs of the semigroup On (A), which is a semigroup of order-preserving and A-decreasing transformations on Xn, are investigated. Let X be a non-empty set and TX be the full transformation semigroup on X. TX istheset ofall functions from X toX andfunctionsareasemigroup with respect to the composition operation. Let n ∈ Z+ and Xn = {1,2,...,n}. In this study we use Tn instead of TXn for convenience. On = {α ∈ Tn : ∀x,y ∈ Xn,x ≤ y ⇐⇒ xα ≤ yα } is called the semigroup of order-preserving transformations on Xn and this structure is a subsemigroup of the semigroup Tn. Let ∅ ̸= A ⊆ Xn and On(A) = {α ∈ On : xα ≤ x (∀x ∈ A)}. Then On(A) is called the semigroup of order-preserving and A-decreasing transformations on Xn, and this semigroup is a subsemigroup of the semigroup On. In this study, the undirected zero divisor graph of the semigroup On(A) is defined and denoted by Γ. It is proven that the Γ is a connected for n ≥ 4, the cardinality of vertices, vertex degrees, the maximum and the minimum vertex degrees, the diameter, the girth of the Γ graph are calculated and a lower bound for the number of cliques is found.