Ordered blueprint associated with a tract

Core idea

Let $F=(G,N_F)$ be a tract. Adjoin an absorbing zero to $G$ , take the free ambient semiring $\mathbb N[G]$ , and generate a partial order by

0\leq\sum_i g_i \qquad\text{for every }\sum_i g_i\in N_F.

Together with the monomial set $G\sqcup\{0\}$ , this defines an ordered blueprint $F^{\mathrm{oblpr}}$ .

Adjunction and its restriction

The construction lands in the idylls. In the reverse direction, an idyll determines a tract from its multiplicative unit group and the formal sums satisfying $0\leq\sum_i g_i$ . These constructions form an adjunction between tracts and idylls.

They are not an equivalence between all tracts and all ordered blueprints. The adjunction restricts to an equivalence between idylls and the tracts whose null sets are ideals in their ambient semirings. Outside that restricted essential image, forming the generated order can add consequences not present in the original tract null set.

References

Matthew Baker and Oliver Lorscheid, “The moduli space of matroids,” Advances in Mathematics 390 (2021), 107883, Theorem 2.21. arXiv:1809.03542.