  The main purpose of this article is to initiate a systematic study of
Semihypergroups, first introduced by C. Dunkl [4], I. Jewett [13] and R.
Spector [20] independently around 1972. We introduce and study several natural
algebraic and analytic structures on semihypergroups, which are well-known in
the case of topological groups and semigroups. In particular, we first study
almost periodic and weakly almost periodic function spaces (basic properties,
their relation to the compactness of the underlying space, introversion and
Arens product on their duals among others). We then introduce homomorphisms and
ideals, and thereby examine their behaviour (basic properties, structure of the
kernel and relation of amenability to minimal ideals) in order to gain insight
into the structure of a Semihypergroup itself. In the process, we further
investigate where and why this theory deviates from the classical theory of
semigroups.
