发表机构
Eszterházy Károly Catholic University; University of Pécs（埃斯泰尔戈姆天主教大学; 佩奇大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文为局部单位对齐有序半群提出典范分解传输理论，通过正幂等骨架凝聚与传输同态族重构乘法与序，并给出序恢复条件，证明全序情形可完全重构。
AI 中文摘要
本文发展了局部单位对齐有序半群的典范分解与重构理论。引入了正幂等骨架的两种凝聚化。较细的一种记录了局部单位块内被迫产生的闭包，而乘法凝聚商则产生了一个典范块的并半格。分量就是这些块的纤维。对于可比较的块$A\le B$，每个正幂等元$q\in B$定义了一个从$A$上的分量到$B$上的分量的传输同态$x\mapsto xq$。保留所有这些映射就得到了一个分解传输族，它取代了普通直接系统中使用的单一连接映射。这些典范数据在无需额外假设的情况下重构了乘法：两个元素被传输到它们的并分量，它们的乘积是相应传输像的最小乘积。它们还决定了从较低分量到较高分量的每一个比较方向。在三个显式序恢复条件中的任何一个下，完整的环境序都被恢复。特别地，每个全序局部单位对齐半群都由其分解传输数据完全重构。一个补充结果在存在合适的保持分量的反自同构时，从分量序对偶性恢复序。当每个接收适当转移的分量具有最小正幂等元时，分解族坍缩为单一的最小目标映射。在自然单调性条件下，这些映射构成一个普通直接系统并诱导一个定向字典序。例子既说明了为什么需要额外的序条件，也说明了为什么分解传输通常不能由普通转移映射替代。
英文摘要
This paper develops a canonical decomposition and reconstruction theory for local-unit-aligned ordered semigroups. Two coherentizations of the positive-idempotent skeleton are introduced. The finer one records the closure forced within local-unit blocks, while the multiplication-coherent quotient yields a join-semilattice of canonical blocks. Components are the fibers of these blocks. For comparable blocks $A\le B$, each positive idempotent $q\in B$ defines a transport homomorphism $x\mapsto xq$ from the component over $A$ to that over $B$. Keeping all such maps gives a resolved-transport family, which replaces the single connecting map used in an ordinary direct system. These canonical data reconstruct multiplication without additional assumptions: two elements are transported to their join component, and their product is the least product of corresponding transported images. They also determine every comparison directed from a lower component to a higher one. The full ambient order is recovered under any of three explicit order-recovery conditions. In particular, every totally ordered local-unit-aligned semigroup is completely reconstructed by its resolved-transport data. A complementary result recovers the order from componentwise order duality when a suitable component-preserving anti-automorphism is available. When each component receiving a proper transition has a least positive idempotent, the resolved family collapses to a single least-target map. Under a natural monotonicity condition, these maps form an ordinary direct system and induce a directed lexicographic order. Examples show both why the additional order conditions are needed and why resolved transports cannot in general be replaced by ordinary transition maps.
