The Representation Of Finite Semigroup

In this paper, the subject which we reserch is the ?nite noncommutative semigroupS which contains 1. And the algebraic semigroup of S is semisimple. we ?rst analyzes theelements of ?nite semigroups in the nature of the elements in the ?nite semigroups s isdivided into four classes.(1)?M∈N, such thats^M = 1;(2)?M∈N, such thats^M+1 = s;(3)?M∈N, t∈N, t < M, such thats^M+1 = s^t;(4)?M∈N, such thats^M+1 = s^M.When the finite semigroups S only contains elements of the first two categories, thesemigroup can be written as disjoint union of a finite number of groups. Next we give therepresentation of a finite semigroup by using the representation of a group, and extendthe irreducible representation,character,character table theorem of a finite group to afinite semigroup. This thesis also provides the representation of the semigroup S when itonly contains the third or the fourth class.