Generalized transformation semigroups and restricted range

The generalized transformation semigroup GT(X,Y;θ) and the semigroup of transformation with restricted range T(X,Y), have been much studied over the last 60 years. Many of algebraic questions were answered over there, as we will see, such as regularity, Green's relations, isomorphism and embedding theorems, generating systems, ranks and some others. One of the important questions which is still unanswered until now is: Determining a minimal generating set and the rank of GT(X,Y;θ), and this was the motivation for us to do more investigations. In this thesis, we give our contributions in this direction. First, we show that GT(X,Y;θ) can be embedded in Τ(Z,Y)= {α∈Τ(Z): Zα ⊆Y } where Z=X∪Y . We also show, if θ: Y→X is one to one, then GT(X,Y;θ) and Τ(X, im(θ)) are isomorphic semigroups. After that, we present our main results as follows: If |X|=m, |Y|=n, and |im(θ)|=r, where m, n, r∈N, then we show that: rank(GT(X,Y;θ))={█(∑_(k=r+1)^(min⁡(m,n))▒〖n!S(m,k)/(n-k)! 〗 if θ is neither onto nor 1-1@ @ @(n¦m) if θ is onto but not 1-1 @ @ @ S(m,r) if θ is 1-1 and 1≤r≤m-1)┤ .