  Expansion was introduced at the end of the 1970s for calculating principal
typings for $\lambda$-terms in intersection type systems. Expansion variables
(E-variables) were introduced at the end of the 1990s to simplify and help
mechanise expansion. Recently, E-variables have been further simplified and
generalised to also allow calculating other type operators than just
intersection. There has been much work on semantics for intersection type
systems, but only one such work on intersection type systems with E-variables.
That work established that building a semantics for E-variables is very
challenging. Because it is unclear how to devise a space of meanings for
E-variables, that work developed instead a space of meanings for types that is
hierarchical in the sense of having many degrees (denoted by indexes). However,
although the indexed calculus helped identify the serious problems of giving a
semantics for expansion variables, the sound realisability semantics was only
complete when one single E-variable is used and furthermore, the universal type
$\omega$ was not allowed. In this paper, we are able to overcome these
challenges. We develop a realisability semantics where we allow an arbitrary
(possibly infinite) number of expansion variables and where $\omega$ is
present. We show the soundness and completeness of our proposed semantics.
