  An algebraic quantum group is a multiplier Hopf algebra with integrals. In
this paper we will develop a theory of algebraic quantum hypergroups. It is
very similar to the theory of algebraic quantum groups, except that the
comultiplication is no longer assumed to be a homomorphism. We still require
the existence of a left and of a right integral. There is also an antipode but
it is characterized in terms of these integrals. We construct the dual, just as
in the case of algebraic quantum groups and we show that the dual of the dual
is the original quantum hypergroup. We define algebraic quantum hypergroups of
compact type and discrete type and we show that these types are dual to each
other. The algebraic quantum hypergroups of compact type are essentially the
algebraic ingredients of the compact quantum hypergroups as introduced and
studied (in an operator algebraic context) by Chapovsky and Vainerman.
  We will give some basic examples in order to illustrate different aspects of
the theory. In a separate note, we will consider more special cases and more
complicated examples. In particular, in that note, we will give a general
construction procedure and show how known examples of these algebraic quantum
hypergroups fit into this framework.
