含反定义域限制的签名下的部分函数完全表示
Complete representation by partial functions for signatures containing antidomain restriction
AI总结：
本研究针对含反定义域限制的多类签名，分析部分函数的并/交完全表示的等价性、可表示的充要条件，证明相关代数类无法被存在-全称-存在一阶理论公理化，并为14类签名构造显式表示，证明其可由全称-存在-全称一阶语句公理化。
摘要：
我们研究了部分函数的完全表示概念，其签名中的运算包含反定义域限制，还可包含复合、交、更新、优先并、定义域、反定义域和差集运算。当签名同时包含反定义域限制和交运算时，并完全表示与交完全表示是一致的。否则，对于我们所研究的签名，交完全表示严格强于并完全表示。代数结构可被交完全表示的一个必要条件是其原子具有分离性。对于我们研究的签名，该条件为充分条件当且仅当签名中不包含复合运算。对于我们研究的每一类签名，（交）完全可表示代数构成的类都无法被任何存在-全称-存在一阶理论公理化。针对14种表达能力不同的签名，我们通过构造显式表示证明，（交）完全可表示代数构成一个基本初等类，可由一条全称-存在-全称一阶语句公理化。我们完成公理化的签名包括：含反定义域限制且可任意包含交、更新、优先并的签名；以及含反定义域限制、复合、交，且可任意包含更新、优先并、定义域、反定义域的签名。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
