Duality for dyadic intervals

Abstract

In an earlier paper, Romanowska, Ślusarski and Smith described a duality between the category of (real) polytopes (finitely generated real convex sets considered as barycentric algebras) and a certain category of intersections of hypercubes, considered as barycentric algebras with additional constant operations. This paper is a first step in finding a duality for dyadic polytopes, analogues of real convex polytopes, but defined over the ring [Formula: see text] of dyadic rational numbers instead of the ring of reals. A dyadic [Formula: see text]-dimensional polytope is the intersection with the dyadic space [Formula: see text] of an [Formula: see text]-dimensional real polytope whose vertices lie in the dyadic space. The one-dimensional analogues are dyadic intervals. Algebraically, dyadic polytopes carry the structure of a commutative, entropic and idempotent groupoid under the operation of arithmetic mean. Such dyadic polytopes do not preserve all properties of real polytopes. In particular, there are infinitely many (pairwise non-isomorphic) dyadic intervals. We first show that finitely generated subgroupoids of the groupoid [Formula: see text] are all isomorphic to dyadic intervals. Then, we describe a duality for the class of dyadic intervals. The duality is given by an infinite dualizing (schizophrenic) object, the dyadic unit interval. The dual spaces are certain subgroupoids of the square of the dyadic unit interval with additional constant operations. A second paper deals with a duality for dyadic triangles.

Duality for dyadic triangles

“…By [8,Lemma 5.7], each non-constant homomorphism h from a dyadic triangle T to D extends uniquely to an affine D-space homomorphismh from the affine D-space D 2 to D. The homomorphismh is determined by the slope of a family of parallel (real) lines. On the other hand, each affine D-space homomorphism from D 2 to D restricts to a groupoid homomorphism from T to D. Since each dyadic triangle is finitely generated (see [10]), it follows that its image h(T ) is also finitely generated, and hence (by results of [8]) is isomorphic to a dyadic interval. The image h(T ) is a subgroupoid of the interval [l, u], where l and u are images of two of the three vertices of T , with l being the lower end and u the upper end of h(T ).…”

“…The paper [8] provided a characterization of finitely generated subgroupoids of the groupoid (D, •) of dyadic rational numbers with the arithmetic mean as the basic operation, and a description of a duality between the category of dyadic intervals and the category of certain subgroupoids of the dyadic unit square. In this paper we extend the earlier results to the case of dyadic triangles, again considered as groupoids, subgroupoids of the groupoid (D 2 , •).…”

“…The notation and results from [8] will be carried over here. In particular, we will use basic facts concerning affine spaces and barycentric algebras described there, as well as basic facts concerning dualities given by dualizing objects.…”

“…In particular, we will use basic facts concerning affine spaces and barycentric algebras described there, as well as basic facts concerning dualities given by dualizing objects. We will frequently use the fact proved in [8], that each finitely generated subgroupoid of (D, •) is isomorphic to a dyadic interval of the form D k = [0, k], where k is an odd positive integer. Such an interval is said to be of type k.…”

“…As in the first paper, we use notation and conventions similar to those of [14,15] and the earlier papers. The reader should consult the first paper [8] for further background and all notions not explicitly defined here. For further details and information on affine spaces and barycentric algebras, we also refer the reader to the monographs [13,14] and to earlier papers.…”