Idyll - Knowlpedia

Definition

An idyll is a band $B$ such that

B^\times=B\setminus\{0\}\ne\varnothing.

Thus every nonzero element of its pointed multiplicative monoid is a unit. The condition is multiplicative: the additive information remains encoded by the nullset $N_B$ .

Fields, hyperfields, and partial fields give examples of idylls. An arbitrary idyll need not be a hyperfield, and its nullset need not be generated by three-term relations.

Associated tract

Because $B^\times=B\setminus\{0\}$ , retaining the unit group and its null formal sums gives an associated tract. Not every tract comes from an idyll: a tract's null set need not be an ideal of the full semiring of formal sums.

Equivalent realization

The ordered-blueprint realization of an idyll is a purely positive ordered blue field with unique weak inverses and freely generated ambient semiring. This is an equivalent presentation of the band definition, not an additional axiom here.

An idyll whose nullset is the fusion ideal generated by its three-term null relations is a pasture.

References

Matthew Baker, Tong Jin, and Oliver Lorscheid, New building blocks for $\mathbb F_1$ -geometry: bands and band schemes, Definition 1.5.