Is there a standard notation that expresses substructure? The specific case that I care about is the following:

Suppose $\sigma,\tau$ are permutations such that $$\sigma(x)\not=x\implies \sigma(x)=\tau(x) \qquad (1)$$ This is equivalent to saying that the cycles appearing in the cycle decomposition of $\sigma$ are a subset of the cycles appearing in the cycle decomposition of $\tau$. Since I am not aware of a special notation to describe this condition, and I use this type of condition frequently in a paper I am working on, I have been using $\sigma\subset\tau$ as a shorthand.

Another way I have been thinking of describing this property is via a projection. For instance, $\sigma\subset\tau$ is the same as saying $\sigma=\tau|_{S'}$ where $S'$ is a suitable embedding of a smaller symmetric group.

When condition $(1)$ is used, it seems like the proper generalization is some sort of "refinement" or "continuation" condition, like saying $\text{supp}(f)\subset\text{supp}(g)$ and $f=g$ on the common support.

So there are three ways of thinking about this property:

- Information about a decomposition
- Restriction to a subspace
- Continuation of a function

Is there a common notation that is used in these three cases? Are there conditions for other kinds of objects that also have interpretations along all these lines?

Just looking for some insight.