Finitely generated dyadic convex sets (2403.17028v1)

Abstract: Dyadic rationals are rationals whose denominator is a power of $2$. We define dyadic $n$-dimensional convex sets as the intersections with $n$-dimensional dyadic space of an $n$-dimensional real convex set. Such a dyadic convex set is said to be a dyadic $n$-dimensional polytope if the real convex set is a polytope whose vertices lie in the dyadic space. Dyadic convex sets are described as subreducts (subalgebras of reducts) of certain faithful affine spaces over the ring of dyadic numbers, or equivalently as commutative, entropic and idempotent groupoids under the binary operation of arithmetic mean. The paper contains two main results. First, it is proved that, while all dyadic polytopes are finitely generated, only dyadic simplices are generated by their vertices. This answers a question formulated in an earlier paper. Then, a characterization of finitely generated subgroupoids of dyadic convex sets is provided, and it is shown how to use the characterization to determine the minimal number of generators of certain convex subsets of the dyadic plane.