AI 中文总结
该研究引入射影逻辑几何，证明两个代数结构强双可解释的条件是其射影逻辑集范畴或射影可定义集范畴相对于特定函子类等价，推广了相关概念，用范畴方法为解释理论带来新视角并建立基本结果。
AI 中文摘要
我们引入射影逻辑几何，并证明两个代数结构是强双可解释的，当且仅当它们的射影逻辑集范畴相对于解释函子类是等价的，这也等同于它们的射影可定义集范畴相对于平移函子类是等价的。这些构造推广了鲍里斯·普洛特金的两个想法：泛代数几何中的几何等价概念以及从泛代数几何到逻辑几何的转变。此外，我们的范畴方法为解释理论提供了新视角，使我们能用范畴方法建立一系列基本结果。
英文摘要
We introduce projective logical geometry and prove that two algebraic structures are strongly bi-interpretable if and only if their categories of projective logical sets are equivalent relative to the class of interpretation functors, which is also equivalent to their categories of projective definable sets being equivalent relative to the class of translation functors. These constructions generalize two ideas of Boris Plotkin: the concept of geometric equivalence in universal algebraic geometry and the transition from universal algebraic geometry to logical geometry. Furthermore, our categorical approach offers a fresh perspective on the theory of interpretations, enabling us to establish a series of fundamental results using categorical methods.
