Additive Ternary $\Gamma$-Modules and Homological Algebra

TL;DR

This paper introduces a new framework for additive ternary modules over commutative monoids, constructs an associated operator ring, and develops homological algebra tools like Ext and Tor within this setting.

Contribution

It defines additive ternary $ ext{Gamma}$-modules, constructs their operator rings, and develops a homological algebra theory including tensor products, Ext, and Tor functors.

Findings

The category of ternary modules is equivalent to modules over an operator ring, making it abelian.

Under unital conditions, the category has enough projectives and injectives.

Explicit computations of Ext and Tor for $ ext{Z}/4 ext{Z}$ illustrate the theory's concrete applications.

Abstract

Fix a commutative monoid $(T,+,0)$, a commutative monoid $(\Gamma,+,0_\Gamma)$, and a map \[ (a,\alpha,b,\beta,c)\longmapsto a\,\alpha\,b\,\beta\,c\in T \] which is additive in each variable and associative in the ternary sense. A left additive ternary $\Gamma$-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} $\mathcal O_{T,\Gamma}$ generated by the left translations $(a,\alpha,b,\beta)$. It is shown that $T\!\!\;\!-\!\Gamma\mathrm{Mod}$ is equivalent to the ordinary module category $\mathcal O_{T,\Gamma}\mathrm{Mod}$. The category is therefore abelian. Under the unital operator hypothesis \emph{(U)} (i.e. when $\mathcal O_{T,\Gamma}$ 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 $\mathrm{Ext}$ and $\mathrm{Tor}$ are developed inside the additive track. A finite example with $T=\ZZ/4\ZZ$ gives explicit computations \[\mathrm{Ext}^1_T(\ZZ/2\ZZ,\ZZ/2\ZZ)\cong \ZZ/2\ZZ,\qquad \mathrm{Tor}^T_1(\ZZ/2\ZZ,\ZZ/2\ZZ)\cong \ZZ/2\ZZ,\] with a concrete nonsplit extension and an explicit failure of tensor exactness. Counterexamples isolate the precise points where naive binary-module arguments break.