Ordinal spaces (Q2300118)

In this paper, an axiomatic approach is proposed for ordinal spaces. According to the authors, an ordinal space is an ordered triplet \((X,L,\delta)\) where \(X\) is a nonempty set, \(L\) is a linearly ordered set with minimal element \(0\), and \(\delta:X\times X\to L\) is a surjective map with\par i) \(\delta(x,y)=\delta(y,x)\), for all \(x,y\in X\)\par ii) \(\delta(x,y)=0\) iff \(x=y\).\par The paper consists of 9 sections, that may be described as follows. Section 1 is introductory. In Section 2, two approaches are provided for defining isomorphisms between ordinal spaces. Section 3 is devoted to the definition of sets of balls in ordinal spaces by starting from the cuts of linearly ordered sets. In Section 4, the topological properties of ordinal spaces are being investigated. Further, in Section 5, an isomorphism criterion is established for the Hasse diagrams \({\mathcal{H}}(B_X)\) and \({\mathcal{H}}(B_Y)\) of two finite ordinal spaces \(X\) and \(Y\). Section 6 is devoted to the formulation of two conjectures about the maximal and minimal number of balls in finite ordinal spaces. The objective of Section 7 and Section 8 is to study the embeddings of ordinal spaces into one dimensional and higher dimensional Euclidean spaces, respectively. Finally, in Section 9, an analog of Gromov-Hausdorff distance is proposed for ordinal spaces with a fixed finite number of points.