Zero divisor graph of order preserving and order decreasing contraction mappings semigroup

In this thesis, the semigroup ODCT n which consists of transformations on the finite set Xn = {1, 2, . . . , n} that are simultaneously orderpreserving, orderdecreasing, and full contraction mappings, is investigated. The main objective of the study is to analyze the zerodivisor structure of this semigroup and to determine the fundamental properties of the corresponding zerodivisor graph Γ(ODCT n). Firstly, the left, the right, and the twosided zerodivisors of the semigroup ODCT n were characterized. It is shown that the number of elements of the left zero divisor set of the ODCT n semigroup is 2n−1 − 1. The number of elements of the right zero divisor set and the two-way zero divisor set are equal to each other and 2n−2. Subsequently, the zerodivisor graph Γ(ODCT n) was defined for n ≥ 3. The zero element of the semigroup ODCT n is denoted by θ. It is shown that the graph is connected for n ≥ 4 and has a diameter of 2. Furthermore, it was mentioned that for n ≥ 5, the girth of the graph is 3. A lower bound for the clique number of the graph was given. This study reveals the relationship between the algebraic structure of the ODCT n semigroup and the topological properties of its zerodivisor graph, contributing to the existing literature. The results obtained provide a foundation for further investigation into other invariants of the graph, such as its diameter, vertex degrees, clique number, and chromatic number.