Band - Knowlpedia

Let $B$ be a commutative monoid with identity $1$ and an absorbing element $0$ , and let

B^+=\mathbb N[B]/\langle 0\rangle

be its semiring of finite formal sums, with the monoid zero identified with the empty sum. Such a monoid is called pointed. A band is a pair $(B,N_B)$ in which the nullset $N_B\subseteq B^+$ is an ideal and, for every $a\in B$ , there is a unique element $-a\in B$ such that

a+(-a)\in N_B.

A morphism of bands $f:B\to C$ is a multiplicative map preserving $0$ and $1$ such that

\sum a_i\in N_B\quad\Longrightarrow\quad\sum f(a_i)\in N_C.

Scope

The symbols in $B^+$ are formal sums. A band does not in general have a single-valued or multivalued addition on $B$ . Rings, hyperrings, and partial fields give bands by recording which formal sums are null, but not every band comes from one of those structures.

References

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