  The infinitary lambda calculi pioneered by Kennaway et al. extend the basic
lambda calculus by metric completion to infinite terms and reductions.
Depending on the chosen metric, the resulting infinitary calculi exhibit
different notions of strictness. To obtain infinitary normalisation and
infinitary confluence properties for these calculi, Kennaway et al. extend
$\beta$-reduction with infinitely many `$\bot$-rules', which contract
meaningless terms directly to $\bot$. Three of the resulting B\"ohm reduction
calculi have unique infinitary normal forms corresponding to B\"ohm-like trees.
  In this paper we develop a corresponding theory of infinitary lambda calculi
based on ideal completion instead of metric completion. We show that each of
our calculi conservatively extends the corresponding metric-based calculus.
Three of our calculi are infinitarily normalising and confluent; their unique
infinitary normal forms are exactly the B\"ohm-like trees of the corresponding
metric-based calculi. Our calculi dispense with the infinitely many
$\bot$-rules of the metric-based calculi. The fully non-strict calculus (called
$111$) consists of only $\beta$-reduction, while the other two calculi (called
$001$ and $101$) require two additional rules that precisely state their
strictness properties: $\lambda x.\bot \to \bot$ (for $001$) and $\bot\,M \to
\bot$ (for $001$ and $101$).
