The Representation Type of the Full Transforamtion Semigroup T ₄

T_n

be the semigroup of all transformations of a set of n elements and k a field of characteristic 0. According to Ponizovskii, the semigroup algebra kT _n is of finite representation type if n≤ 3. According to Putcha, kT _n is of infinite representation type if n≥ 5. Here, we deal with the remaining case n=4 and show that kT ₄ is also of finite representation type. Note that the quiver of kT ₄ already has been exhibited by Putcha, here we determine the relations. It turns out that kT ₄ is a string algebra and its global dimension is 3.