三元$\Gamma$-半模中的Barr正合性与基于同余的同调代数
Barr-Exactness and Congruence-Based Homological Algebra in Ternary $Γ$-Semimodules
AI总结：
本文在三元$\Gamma$-半模范畴中建立同余优先的同调代数演算，定义同余意义下的核与商，证明态射典范分解与同余版同构定理，给出Barr-正合表述，并得到投射生成准则与经典图引理失效的结论。
AI中文摘要：
设$T$为三元$\Gamma$-半环，$\mathsf{SMod}_{T,\Gamma}$为具有内在五元作用$T\times \Gamma\times M \times \Gamma\times T\to M$的三元$\Gamma$-半模范畴。本文在$\mathsf{SMod}_{T,\Gamma}$中建立了同余优先的演算体系。核与商均采用同余意义下的定义，而非子对象的陪集定义，原因是在没有减法运算的情况下陪集商通常不成立。我们证明每个态射都可通过其核同余得到典范分解，建立了三个同构定理的同余版本，并将余等化子描述为由生成同余诱导的商。正合性按正则/Barr-正合的意义定义：核为核对，正合性通过核与正则像的相等来表述。最后，我们记录了该Barr-正合但非阿贝尔的框架下的一个投射生成准则，以及一个经典图引理的具体失效情况。
英文摘要：
Let $T$ be a ternary $Γ$--semiring and let $\mathsf{SMod}_{T,Γ}$ be the category of ternary $Γ$--semimodules with the intrinsic five--ary action $T\times Γ\times M \times Γ\times T\to M$.This paper develops a congruence--first calculus in $\mathsf{SMod}_{T,Γ}$.Kernels and quotients are taken in the sense of congruences, not in the sense of cosets of subobjects, because coset quotients generally fail without subtraction. We prove that every morphism admits a canonical factorization through its kernel congruence, establish congruence versions of the three isomorphism theorems, and describe coequalizers as quotients by generated congruences. Exactness is formulated in the regular/Barr-exact sense: kernels are kernel pairs and exactness is expressed by equality of kernels with regular images. Finally, we record a projective generation criterion and a concrete failure of a classical diagram lemma in this Barr-exact but non-abelian setting.
