  We investigate notions of complete representation by partial functions, where
the operations in the signature include antidomain restriction and may include
composition, intersection, update, preferential union, domain, antidomain, and
set difference. When the signature includes both antidomain restriction and
intersection, the join-complete and the meet-complete representations coincide.
Otherwise, for the signatures we consider, meet-complete is strictly stronger
than join-complete. A necessary condition to be meet-completely representable
is that the atoms are separating. For the signatures we consider, this
condition is sufficient if and only if composition is not in the signature. For
each of the signatures we consider, the class of (meet-)completely
representable algebras is not axiomatisable by any
existential-universal-existential first-order theory. For 14 expressively
distinct signatures, we show, by giving an explicit representation, that the
(meet-)completely representable algebras form a basic elementary class,
axiomatisable by a universal-existential-universal first-order sentence. The
signatures we axiomatise are those containing antidomain restriction and any of
intersection, update, and preferential union and also those containing
antidomain restriction, composition, and intersection and any of update,
preferential union, domain, and antidomain.
