Barr-Exactness and Congruence-Based Homological Algebra in Ternary $Γ$-Semimodules
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.