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

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

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

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

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

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

$w_ i | a_{ij}$.

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