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Definition
A band whose nullset is closed under the fusion rule.
A fusion band is a band whose nullset satisfies the fusion rule:
for all and .
Equivalently, is a fusion ideal of . This condition specifies how existing null relations compose; it does not say that the nullset is generated by relations of any particular arity.
Matthew Baker, Tong Jin, and Oliver Lorscheid, New building blocks for -geometry: bands and band schemes, Definition 1.6.
Let be a commutative monoid with identity and an absorbing element , and let
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 in which the nullset is an ideal and, for every , there is a unique element such that
A morphism of bands is a multiplicative map preserving and such that
For a band , its nullset is the distinguished ideal
Thus the empty sum lies in , and
Membership means that the formal sum is declared null.
The band axiom additionally requires each to have a unique with . Ideal closure alone does not imply the fusion rule.