I am reading about relations between equations using kronecker products as well as the use of ⊕ in situations such as M1 ⊕ M2, M ⊕ N, S ⊕ S, S⊕S⊕…n and others besides. In these papers they are called composite systems.
I understand that in a kronecker product a multiplication is performed on each element of the first matrix by every element in the second and it forms a block matrix of the two systems. Such as that Amxn ⊗ Bpxq is the matrix pm x nq. With situations such as a11 x b11, a11xb12, etc...
But what kind of equation is the followed out by the ⊕ symbol in the case of the above equations?
You are working in the category of vector spaces over some field $k$. Let us denote that category by $C$.
If $M,N\in \text{obj}(C)$ then $M\oplus N$ denotes the direct sum of spaces and $M\otimes N$ denotes the tensor product of two spaces. These are both objects of this category.
Suppose now that you have two morphisms $f_1:M_1\to N_1$ and $f_2:M_2\to N_2$. The direct sum will give you the direct sum of morphisms $f_1\oplus f_2:(M_1\oplus M_2)\to (N_1\oplus N_2)$ and the tensor product will give a morphism $f_1\otimes f_2:(M_1\otimes M_2)\to (N_1\otimes N_2)$.
Since $f_1\oplus f_2$ and $f_1\otimes f_2$ are morphisms in the category $C$ it means they can be represented by matrix multiplication. If $A$ is the matrix that represents $f_1$ and $B$ is the matrix that represents $f_2$ then the "direct sum of matrices" is just the corresponding matrix for $f_1\oplus f_2$ which is formed by sticking the two matrices diagonally. Likewise for $f_1\otimes f_2$ you get the "Kronecker product" for the two matrices.