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')