SAMS Subject Classification Number: 29

We consider maps of the form $A\otimes B\to A$ (respectively $A\otimes B\to B$), induced by a map $B\to 1$ (respectively $A\to 1$). In the case where $A$ and $B$ are spaces of square matrices, these maps are called block (respectively partial) operations. This talk will characterise some classes of block and partial operations, with the main focus on their role in the theory of Kronecker quotients and tensor decompositions in terms of quotients. In this talk we will primarily consider two types of Kronecker quotients, namely linear quotients which are described in terms of the block and partial trace, and multiplicative quotients which are described in terms of determinant-like operations.