I finished working on algebraic cauchy sequences in june and have since then been having fun with universal algebra and generalizing algebraic structures and concepts and I had a thought today. So I am posting this as a question to see if it is valid and if there are any good sources to read up on it.
In classical abstract algebra of rings, groups and modules we have the following definition, I will use group as it is in most cases based on it.
For a given filtration $(G_n)$ of normal subgroups of $G$ such that $G_{n-1}\geq G_n$ we have that a sequence $(x_i)$ is cauchy if for a given $K$ that there exists an $N$ such that for $n,m\geq N$ we have that $x_nG_k=x_mG_k$.
All dandy and fine for those categories but I was thinking if we expand it to be more general, universally so, what is needed? Obviously dealing with subgroups, ideals and such cannot be done, not all algebraic structures have those or something equivalent and even if they got something similar the differens in axioms and structure can still lend to anomolies appearing that makes it cumbersome to work with.
So the obvious thing to go with is Congruence relatins then. So instead of a filtration of normal subgroups or the likes we would have this.
For a given filtration $(\mathcal{R}_n)$ of congruence relations, $\mathcal{R}_n\in\text{Con}(A)$, such that $\mathcal{R}_n\subseteq\mathcal{R}_{n-1}$.
That should take care of that issue as congruence relations are defined to respect the given structure we wouldn't need to worry about anything. And the subset part ensures it behaves like the other did. For the actual sequence part it'd still remain very similar and as $xN$ for a normal subgroup is just the equivalence class of $x \mod N$ we can go for equivalence classes such that it becomes
A sequence $(x_i)\in A$ is cauchy if for a given $K$ there exists an $N$ such that for $n,m\geq N$ we have that $x_n/\mathcal{R}_K=x_m/\mathcal{R}_K$
Which should be a workable translation of those speical cases in the begining to a universal one that applies to any algebraic structure. With pointwise operations it would form another one of those structures as well as congruence relations makes it certain it will work.
Edit - Addition: For completion we have also this definition.
Let $\mathcal{C}$ be the collection of cauchy sequences. Then a sequence $(x_i)$ is a null sequence if for every $K$ we have an $N$ such that $i>N$ means that $x_i\in G_k$. Mark this subgroup as $N$, then $\mathcal{C}/N$ is the completion of the group with respect to the filtration.
I thought further ahead on that issue and I attempt to translate it into this statement.
Let $\mathcal{C}$ be the collection of cauchy sequences. Then two sequences $(x_i)$ and $(y_i)$ are in the relation $\mathcal{R}$ if for a given $K$ we have there exists an $N$ such that for $i>N$ that $x_i/\mathcal{R}_K =y_i/\mathcal{R}_K$. Then the completion is then $\mathcal{C}/\mathcal{R}$.
This is clearly a congruence relation and as such the quotient is meaningful and works.
Is this a proper translation of the definition to the language of universal algebra? Are there any possible sources that have worked on it?