Nullset of a band - Knowlpedia

For a band $B$ , its nullset is the distinguished ideal

N_B\subseteq B^+=\mathbb N[B]/\langle0\rangle.

Thus the empty sum lies in $N_B$ , and

\alpha,\beta\in N_B,\quad \gamma\in B^+ \quad\Longrightarrow\quad \alpha+\beta\in N_B,\qquad \gamma\alpha\in N_B.

Membership $\sum a_i\in N_B$ means that the formal sum $\sum a_i$ is declared null.

The band axiom additionally requires each $a\in B$ to have a unique $-a\in B$ with $a+(-a)\in N_B$ . Ideal closure alone does not imply the fusion rule.

Terminology

The band literature uses the single word nullset for $N_B$ . This is not the same as a null ideal used to form a quotient of a band: the nullset is part of the band itself.

References

Matthew Baker, Tong Jin, and Oliver Lorscheid, New building blocks for $\mathbb F_1$ -geometry: bands and band schemes, Definition 1.1.