Definition 55.3.1. A numerical type $T$ is given by

$n, m_ i, a_{ij}, w_ i, g_ i$

where $n \geq 1$ is an integer and $m_ i$, $a_{ij}$, $w_ i$, $g_ i$ are integers for $1 \leq i, j \leq n$ subject to the following conditions

1. $m_ i > 0$, $w_ i > 0$, $g_ i \geq 0$,

2. the matrix $A = (a_{ij})$ is symmetric and $a_{ij} \geq 0$ for $i \not= j$,

3. there is no proper nonempty subset $I \subset \{ 1, \ldots , n\}$ such that $a_{ij} = 0$ for $i \in I$, $j \not\in I$,

4. for each $i$ we have $\sum _ j a_{ij}m_ j = 0$, and

5. $w_ i | a_{ij}$.

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