Band as an ordered blueprint

Core idea

Let $B$ be a band with null set $N_B$ . Its associated ordered blueprint is

B^{\mathrm{oblpr}}=(B,B^+,\leq),

where $B$ is the distinguished multiplicative monoid, $B^+$ is its ambient semiring of formal sums, and the order is determined by

0\leq\sum a_i \quad\Longleftrightarrow\quad \sum a_i\in N_B.

A band morphism extends termwise to an order-preserving semiring homomorphism, and this construction defines a fully faithful functor

(-)^{\mathrm{oblpr}}:\mathbf{Bands}\hookrightarrow \mathbf{OBlpr}_{\mathbb F_1^\pm}.

Coreflection

For an ordered blueprint $C$ over $\mathbb F_1^\pm$ , define $C^{\mathrm{band}}$ to have underlying monoid $C^\bullet$ and null set

N_{C^{\mathrm{band}}} = \left\{\sum a_i\ \middle|\ 0\leq\sum a_i\text{ in }C\right\}.

Then $(-)^{\mathrm{band}}$ is right adjoint to $(-)^{\mathrm{oblpr}}$ , with natural bijections

\operatorname{Hom}_{\mathbf{OBlpr}} \bigl(B^{\mathrm{oblpr}},C\bigr) \cong \operatorname{Hom}_{\mathbf{Bands}} \bigl(B,C^{\mathrm{band}}\bigr).

Moreover $(B^{\mathrm{oblpr}})^{\mathrm{band}}=B$ . Thus bands form a coreflective full subcategory of ordered blueprints over $\mathbb F_1^\pm$ .

References