Definition 6

Let $(\Delta_i)_{i \in I}$ be a finite family of commutative monoids written additively. A system of commutation factors over the $\Delta_i$ with values in a commutative ring $\mathbf{A}$ is a system of mappings $\varepsilon_{ij}: \Delta_i \times \Delta_j \to \mathbf{A}$, where $i \in I$, $j \in I$, $i \neq j$ satisfying the following conditions:

(22)

(23) (24)

for all $\alpha_i, \alpha'_i$ in $\Delta_i$, $\beta_j, \beta'_j$ in $\Delta_j$.

If I is given a total ordering and the $\Delta_i$ are groups, a system of commutation factors is defined over the $\Delta_i$ by taking for every ordered pair $(i, j)$ such that $i < j$ an arbitrary $\mathbf{Z}$-*bilinear* mapping of $\Delta_i \times \Delta_j$ into the (multiplicative) $\mathbf{Z}$-module $\mathbf{A}^*$ of invertible elements of the ring $\mathbf{A}$ and then writing

for $i < j$.

Note that, since the $\varepsilon_{ij}(\alpha_i, \beta_j)$ are invertible,

by virtue of (22) and (23).