Fusion rule for bands

Definition

Let $B$ be a band, and let $I\subseteq B^+$ be an ideal of formal sums. The fusion rule is the closure condition

c+\sum_i a_i\in I,\qquad -c+\sum_j b_j\in I \quad\Longrightarrow\quad \sum_i a_i+\sum_j b_j\in I.

An ideal satisfying this rule is a fusion ideal.

For $S\subseteq B^+$ , the notation

\langle\!\langle S\rangle\!\rangle

denotes the smallest fusion ideal containing $S$ . It is obtained by closing $S$ under ideal operations and the displayed fusion rule.

Interpretation

Fusion cancels $c$ against its unique additive inverse $-c$ across two null relations. It is an additional closure axiom: the nullset of an arbitrary band need not be a fusion ideal.

References

Matthew Baker, Tong Jin, and Oliver Lorscheid, New building blocks for $\mathbb F_1$ -geometry: bands and band schemes, Definition 1.6.