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.

翻译：我们研究了偏函数完全表示的概念，其中签名中的操作包括反域限制，并可能包含复合、交、更新、优先并、域、反域和集合差。当签名同时包含反域限制和交时，并完全表示与交完全表示一致。否则，对于我们考虑的签名，交完全表示严格强于并完全表示。成为交完全可表示的一个必要条件是原子具有分离性。对于我们考虑的签名，当且仅当签名中不包含复合操作时，该条件才是充分的。对于我们考虑的每个签名，（交）完全可表示代数的类无法通过任何存在-全称-存在一阶理论公理化。对于14种表达力不同的签名，我们通过给出显式表示证明（交）完全可表示代数构成一个基本初等类，可由一个全称-存在-全称一阶语句公理化。我们公理化的签名包括：包含反域限制及交、更新、优先并中任意操作的签名，以及包含反域限制、复合、交及更新、优先并、域、反域中任意操作的签名。