Does representation theory exists without Groups?

Question

I need to know: is representation theory all about Groups?
Is it necessary to be a finite group?
Does representation theory exists without Groups?
For example is there sample where representation is about semi-group?

Answer 1

Yes, there is a matrix representation theory of semigroups. For a survey, see Representations of semigroups by linear transformations.
The term "representation theory" is much more general, not only for groups. Another well-known example is the representation theory of Lie algebras.

Answer 2

The notion of an association scheme is a generalization of the notion of a group. Many group-theoretic concepts — including representation theory — have natural generalizations to association schemes. The canonical reference is the book Theory of Association Schemes by Paul-Hermann Zieschang.

Answer 3

For a more recent reference, see

J. Almeida, S. W. Margolis, B. Steinberg and M. V. Volkov, Representation theory of finite semigroups, semigroup radicals and formal language theory, Trans. Amer. Math. Soc. 361 (2009), 1429-1461.

This paper contains an extensive bibliography on the representation theory of semigroups.