Additive Ternary $\mathrm{\Gamma }$-Modules and Homological Algebra

Fix a commutative monoid $(T,+,0)$, a commutative monoid $(\mathrm{\Gamma },+,0_{\mathrm{\Gamma }})$, and a map $$(a,\alpha ,b,\beta ,c)⟼a\alpha b\beta c\in T$$ which is additive in each variable and associative in the ternary sense. A left additive ternary $\mathrm{\Gamma }$-module is an abelian group $M$ equipped with an action $(a,\alpha ,b,\beta ,m)\mapsto a\alpha b\beta m$ satisfying the same associativity constraints. Two scalars are intrinsic, so the action is genuinely polyadic; in particular, we do not assume a canonical unary action $T\times M\to M$. The first part constructs the \emph{operator ring} ${\mathcal{O}}_{T,\mathrm{\Gamma }}$ generated by the left translations $(a,\alpha ,b,\beta )$. It is shown that $T\text{\hspace{0.17em}⁣}\text{\hspace{0.17em}⁣}\text{\hspace{0.17em}⁣}-\text{\hspace{0.17em}⁣}\mathrm{\Gamma }\mathrm{M}\mathrm{o}\mathrm{d}$ is equivalent to the ordinary module category ${\mathcal{O}}_{T,\mathrm{\Gamma }}\mathrm{M}\mathrm{o}\mathrm{d}$. The category is therefore abelian. Under the unital operator hypothesis \emph{(U)} (i.e. when ${\mathcal{O}}_{T,\mathrm{\Gamma }}$ is unital), it has enough projectives and enough injectives. The second part defines a tensor product over $T$ by a \emph{bi-balanced} universal property forced by the ternary action, proves right exactness of $-{\otimes }_T-$, and establishes a Tensor–Hom adjunction for bimodules. Assuming \emph{(U)}, derived functors $\mathrm{E}\mathrm{x}\mathrm{t}$ and $\mathrm{T}\mathrm{o}\mathrm{r}$ are developed inside the additive track. A finite example with $T=\text{ZZ}\mathrm{/}4\text{ZZ}$ gives explicit computations $${\mathrm{E}\mathrm{x}\mathrm{t}}_T^1(\text{ZZ}\mathrm{/}2\text{ZZ},\text{ZZ}\mathrm{/}2\text{ZZ})\cong \text{ZZ}\mathrm{/}2\text{ZZ},{\mathrm{T}\mathrm{o}\mathrm{r}}_1^T(\text{ZZ}\mathrm{/}2\text{ZZ},\text{ZZ}\mathrm{/}2\text{ZZ})\cong \text{ZZ}\mathrm{/}2\text{ZZ},$$ with a concrete nonsplit extension and an explicit failure of tensor exactness. Counterexamples isolate the precise points where naive binary-module arguments break.