AI 中文总结
本文基于G. Birkhoff的极性概念，在带0的偏序集上引入正交伴随性，刻画其性质、构造方法及延拓条件，通过示例说明结果。
AI 中文摘要
受G. Birkhoff为集合上的二元关系引入的极性概念启发，我们在带0的偏序集上引入正交性概念。带0的偏序集上的一对算子f、g称为正交伴随，当且仅当f(x)与y正交等价于x与g(y)正交。我们刻画了给定f时g的存在性与唯一性，描述了正交伴随的基本性质，给出了伪补偏序集上正交伴随对的构造方法。若给定算子f是满足某些自然性质的伪补偏序集的序同构，则可明确描述其对应的伴随g；此外，若f与f⁻¹是双射⊥-态射，则它们也互为正交伴随。最后，我们证明了偏序集P上给定的一对正交伴随映射可能无法延拓到P的Dedekind-McNeille完备化，并给出了此类延拓存在的充分条件；还给出了正交伴随映射延拓到理想格的充分条件，我们的结果由大量示例说明。
英文摘要
Motivated by the concept of polarity introduced by G. Birkhoff for a binary relation on a set, we introduce a concept of orthogonality in a poset with $0$. A pair of operators $f$, $g$ on a poset with $0$ is called orthogonally adjoint if $f(x)$ is orthogonal to $y$ if and only if $x$ is orthogonal to $g(y)$. We characterize the existence and the uniqueness of $g$ for given $f$ and describe basic properties of orthogonal adjointness. We present constructions of orthogonally adjoint pairs in pseudocomplemented posets. If a given operator $f$ is an order-isomorphism of a pseudocomplemented poset satisfying some natural properties then the corresponding adjoint $g$ can be described explicitly. Moreover, if $f$ and $f^{-1}$ are bijective $\perp$-morphisms then they are orthogonally adjoint, too. Finally we show that a given pair of orthogonally adjoint mappings on a poset $\mathbf P$ may not be extendable to the Dedekind-McNeille completion of $\mathbf P$ and we present sufficient conditions for the existence of such an extension. We also provide sufficient conditions for the existence of an extension of orthogonally adjoint mappings to the lattice of ideals. Our results are illustrated by numerous examples.
