  It is well-known that any weak Hopf algebra gives rise to a Hopf algebroid.
Moreover it is possible to characterize those Hopf algebroids that arise in
this way.
  Recently, the notion of a weak Hopf algebra has been extended to the case of
algebras without identity. This led to the theory of weak multiplier Hopf
algebras. Similarly also the theory of Hopf algebroids was recently developed
for algebras without identity. They are called multiplier Hopf algebroids. Then
it is quite natural to investigate the expected link between weak multiplier
Hopf algebras and multiplier Hopf algebroids. This relation has been considered
already in the original paper on multiplier Hopf algebroids. In this note, we
investigate the connection further.
  First we show that any regular weak multiplier Hopf algebra gives rise, in a
natural way, to a regular multiplier Hopf algebroid. Secondly we give a
characterization, mainly in terms of the base algebra, for a regular multiplier
Hopf algebroid to have an underlying weak multiplier Hopf algebra. We
illustrate this result with some examples. In particular, we give examples of
multiplier Hopf algebroids that do not arise from a weak multiplier Hopf
algebra.
