Classifications of maximal elements and maximal subsemigroups in semigroups of transformations with invariant set

Let X be a nonempty set and let T(X) denote the full transformation semigroup on X. For a fixed nonempty subset Y of X, we define S(X, Y) as the subsemigroup of T(X) given by S(X, Y) = {ω → T(X) : Y ω ↑ Y }. In this thesis, the first part, we characterize all maximal elements in the semigroup S(X, Y) with respect to the natural partial order. The result in this part corrects an error in the work of Sun and Wang (Sun and Wang, 2013). Moreover, in the second part, we identify all maximal subsemigroups of S(X, Y) when X is a finite set and Y is a proper subset of X. Additionally, we show that within this conditions, the maximal subsemigroups and the maximal regular subsemigroups coincide if and only if |Y| = 1.