Definition
Pasture
An idyll whose nullset is the fusion ideal generated by its three-term null relations.
Definition
A pasture is an idyll whose nullset is the fusion ideal generated by its three-term null relations:
The double brackets denote closure under ideal operations and the fusion rule. Consequently every pasture is a fusion band.
Morphisms are band morphisms: they preserve multiplication, , , and null sums.
Equivalent terminology
A band satisfying the displayed generation condition is called a hereditary fusion band. Hence a pasture is equivalently a hereditary fusion band that is also an idyll. It is not merely an arbitrary ordered blue field or tract.
Fields, partial fields, and hyperfields have associated pastures. The three-term condition is substantive: a general idyll can contain higher-arity null relations not generated from its three-term ones by fusion.
Three-term null relations encode the Grassmann–Plücker relations used for weak matroids and cross-ratios. Fusion then derives the higher-arity null relations by cancelling a term against its additive inverse.
References
Matthew Baker, Tong Jin, and Oliver Lorscheid, New building blocks for -geometry: bands and band schemes, §§1.2.4–1.2.5.