Factorisations in transformation semigroups

In this thesis, studies and results related to factorization in transformation semigroups are investigated and compiled. It has been observed that the classical decomposition of permutations into disjoint cycles can be extended to more general mappings by means of path-cycles, and an algorithm is given to obtain the decomposition. The device is used to obtain information about generating sets for the semigroup of all singular selfmaps of X_n={1,2,…,n}. Let T_(n,r)=S_n∪K_(n,r), where S_n is the symmetric group and K_(n,r) is the set of maps α∶X_n→X_n such that |im(α)|≤r. It is shown that cont(T_(n,r) )=p_r (n) where cont(T_(n,r) )=min{|G|:G⊆K_(n,r) ve 〈G∪S_n 〉=T_(n,r) } 〖,p〗_r (n) is the partition the positive integer n with r terms.