  In this paper, we generalize Majid's bicrossproduct construction. We start
with a pair (A,B) of two regular multiplier Hopf algebras. We assume that B is
a right A-module algebra and that A is a left B-comodule coalgebra. We recall
and discuss the two notions in the first sections of the paper. The right
action of A on B gives rise to the smash product A # B. The left coaction of B
on A gives a possible coproduct on A # B. We will discuss in detail the
necessary compatibility conditions between the action and the coaction for this
to be a proper coproduct on A # B. The result is again a regular multiplier
Hopf algebra. Majid's construction is obtained when we have Hopf algebras.
  We also look at the dual case, constructed from a pair (C,D) of regular
multiplier Hopf algebras where now C is a left D-module algebra while D is a
right C-comodule coalgebra. We will show that indeed, these two constructions
are dual to each other in the sense that a natural pairing of A with C and of B
with D will yield a duality between A # B and the smash product C # D.
  We show that the bicrossproduct of algebraic quantum groups is again an
algebraic quantum group (i.e. a regular multiplier Hopf algebra with
integrals). The *-algebra case will also be considered. Some special cases will
be treated and they will be related with other constructions available in the
literature.
  Finally, the basic example, coming from a (not necessarily finite) group G
with two subgroups H and K such that G=KH and the intersection of H and K is
trivial will be used throughout the paper for motivation and illustration of
the different notions and results. The cases where either H or K is a normal
subgroup will get special attention.
