AI 中文总结
本文在纯组合框架下研究超映射的超边部分对偶，通过中间映射及其修正版本分别给出欧拉性与二部性的交叉刻画，并统一了可定向超映射与带状图的已知结果。
AI 中文摘要
我们在纯组合框架下研究有限超映射的超边部分对偶，不假定可定向性。超映射由其旗集上的三个无不动点对合$（\ au_0,\ au_1,\ au_2）$表示。我们首先给出该模型下中间映射的显式构造：$02$-轨道成为中间顶点圆盘，而$\ au_1$-换位成为中间带；局部定向系统及其扭转数据随后提供中间映射的带符号旋转描述。接下来，我们证明与所选超边集关联的状态圆与相应部分对偶的顶点轨道自然双射，从而得到所有欧拉超边部分对偶的交叉总数刻画。对于二部性，扭转数据导致一个修正的中间映射，其中插入的条记录全局定向的障碍。我们证明部分对偶是二部的当且仅当其被对偶化的超边集恰好是由该修正中间映射的全交叉定向所识别的$c$-型超边集。当超映射可定向时，这些构造特化为已知的可定向超映射结果；当每条超边的价为二时，它们特化为带状图结果。
英文摘要
We study hyperedge partial duals of finite hypermaps in a purely combinatorial framework, without assuming orientability. A hypermap is represented by three fixed-point-free involutions $(τ_0,τ_1,τ_2)$ on its flag set. We first give an explicit construction of the medial map from this model: $02$-orbits become the medial vertex discs, while $τ_1$-transpositions become the medial bands; a local orientation system and its twist data then provide a signed rotation description of the medial map. We next prove that the state circles associated with a chosen set of hyperedges are in natural bijection with the vertex orbits of the corresponding partial dual, yielding a crossing-total characterization of all Eulerian hyperedge partial duals. For bipartiteness, the twist data lead to a modified medial map in which inserted bars record the obstruction to a global orientation. We prove that a partial dual is bipartite if and only if its dualized hyperedge set is exactly the set of $c$-type hyperedges identified by an all-crossing orientation of this modified medial map. When the hypermap is orientable, these constructions specialize to the known orientable-hypermap results; when every hyperedge has valence two, they specialize to the ribbon-graph results.
