Let $E$ be the set of legal entities and $I$ be the set of instruments.

For any set $S$ we define $M(S) = \mathbf{Z}^S = \{\pi\colon S\to\mathbf{Z}\}$ to be the set of all functions from $S$ to $\mathbf{Z}$. $M(S)$ is a module over the ring of integers. Let $\mathcal{L}(M(S))$ be the space of all linear functions from $M(S)$ to $M(S)$.

Instrument amounts are quoted as an integer multiple of their smallest traded unit.

A position is an element of the module $M(I\times E)$.

A transaction occurs when a buyer and a seller exchanging positions.

Given a positions $\pi = a_{i,e}$ and $\pi' = a'_{i',e'}$. define $\xi\colon M\times M\to M\times M$ by $\xi(\pi,\pi') = (a_{i,e'}, a'_{i,e'})$.

$\xi\colon M\times\to M$ by $M - \pi + \xi\pi - \pi' + \xi(\pi')$