赋值镶嵌
Valued Mosaics
AI总结：
该研究为镶嵌配备赋值，构建赋值镶嵌的松弛范畴，分析结合性的因子嵌套条件，为超群相关范畴的性质研究提供新方向。
AI中文摘要：
中村-雷耶斯（Nakamura--Reyes）证明，交换镶嵌（即无结合性的幺正可逆超群）范畴是完备、余完备且具有自由对象的，这与交换多群形成对比。克拉斯纳（Krasner）的多值加法围绕超度量球设计，我们将这种赋值论动机作为核心，为镶嵌配备赋值——即到热带多群$T(\boldsymbol{\u0393})$的保单位幺正态射。保态射构成$T(\boldsymbol{\u0393})$上的切片范畴，该范畴完备且余完备。固定有序阿贝尔群$\boldsymbol{\u0393}$上的更大的赋值镶嵌松弛范畴是有限完备的，且具有所有小余积；对于平凡赋值群，它可恢复交换镶嵌范畴，而非平凡$\boldsymbol{\u0393}$下的松弛余等化子仍待解决。结合性通过因子嵌套分析，该性质在全性下蕴含结合性，且在克拉斯ner超域、$T(\boldsymbol{\u0393})$等全乘积超度量镶嵌中刻画结合性，但无全性时严格更弱。在满足因子嵌套的全赋值镶嵌中，克拉斯纳球公理导出结合性，并升级弱赋值，使得和为超度量球，进而得到上确界典范结构。
英文摘要：
Nakamura--Reyes showed that the category of commutative mosaics---unital reversible hypermagmas, without associativity---is complete, cocomplete, and has free objects, in contrast to commutative polygroups. Krasner's multivalued addition is designed around ultrametric balls; we take that valuation-theoretic motivation as primary and equip mosaics with valuations, as unit-reflecting unitary morphisms into the tropical polygroup $T(Γ)$. Value-preserving morphisms form the slice over $T(Γ)$, which is complete and cocomplete. The larger lax category of valued mosaics over a fixed ordered abelian group $Γ$ is finitely complete and has all small coproducts; it recovers the category of commutative mosaics for the trivial value group, while lax coequalizers for nontrivial $Γ$ remain open. Associativity is analysed via factor nesting, which implies it under totality and characterises it among total product-ultrametric mosaics such as the Krasner hyperfield and $T(Γ)$, yet is strictly weaker without totality. Among total valued mosaics satisfying factor nesting, the Krasner ball axiom yields associativity and upgrades the weak valuation so that sums are ultrametric balls and the superiorly canonical package follows.
