# Structure-preserving layer in a network with respect to a transformation

I'm reading this paper: https://arxiv.org/pdf/1602.07576.pdf. I'll quote the relevant bits:

Deep neural networks produce a sequence of progressively more abstract representations by mapping the input through a series of parameterized functions. In the current generation of neural networks, the representation spaces are usually endowed with very minimal internal structure, such as that of a linear space $$\mathbb{R}$$^n.

In this paper we construct representations that have the structure of a linear $$G$$-space, for some chosen group $$G$$. This means that each vector in the representation space has a pose associated with it, which can be transformed by the elements of some group of transformations $$G$$. This additional structure allows us to model data more efficiently: A filter in a $$G$$-CNN detects co-occurrences of features that have the preferred relative pose [...]

A representation space can obtain its structure from other representation spaces to which it is connected. For this to work, the network or layer $$\phi$$ that maps one representation to another should be structure preserving. For $$G$$-spaces this means that $$\phi$$ has to be equivariant: $$\phi(T_gx)=T'_g\phi(x)$$That is, transforming an input $$x$$ by a transformation $$g$$ (forming $$T_gx$$) and then passing it through the learned map $$\phi$$ should give the same result as first mapping $$x$$ through $$\phi$$ and then transforming the representation.

Equivariance can be realized in many ways, and in particular the operators $$T$$ and $$T'$$ need not be the same. The only requirement for $$T$$ and $$T'$$ is that for any two transformations $$g$$ and $$h$$, we have $$T(gh) = T (g)T (h)$$ (i.e. $$T$$ is a linear representation of $$G$$).

I didn't understand the paragraph in bold. A structure preserving map is something that preserves some operation between elements in the underlying set. A simple example: if $$f:\mathbb{R}^3\to\mathbb{R}$$ such that $$(x,y,z)^T\mapsto x+y+z$$, then $$f(r+s)=f((r_1,r_2,r_3)^T+(s_1,s_2,s_3)^T)=f((r_1+s_1,r_2+s_2,r_3+s_3)) \\=r_1+s_1+r_2+s_2+r_3+s_3=f(r)+f(s)$$ where the addition in the far left term is in $$\mathbb{R}^3$$ and addition in the far right is in $$\mathbb{R}$$. So the map $$f$$ preserves the additional structure of addition.

In the quoted paragraph, $$\phi$$ is the structure preserving map, but what's the structure being preserved exactly? And why is the operator on the right different from one on the left? i.e. $$T'$$ on RHS instead of $$T$$