Abstract
Each of the descriptions of vertices, edges, and facets of the order and chain polytope of a finite partially ordered set are well known. In this paper, we give an explicit description of faces of 2-dimensional simplex in terms of vertices. Namely, it will be proved that an arbitrary triangle in 1-skeleton of the order or chain polytope forms the face of 2-dimensional simplex of each polytope. These results mean a generalization in the case of 2-faces of the characterization known in the case of edges.
Highlights
The combinatorial structure of the order polytope OpPq and the chain polytope C pPq of a finite poset P is explicitly discussed in [1]
It is proved that the number of edges of the order polytope OpPq is equal to that of the chain polytope C pPq in [3]
In the present paper we give an explicit description of faces of 2-dimensional simplex of OpPq and C pPq in terms of vertices
Summary
The combinatorial structure of the order polytope OpPq and the chain polytope C pPq of a finite poset (partially ordered set) P is explicitly discussed in [1]. In [2], the problem when the order polytope OpPq and the chain polytope C pPq are unimodularly equivalent is solved. It is proved that the number of edges of the order polytope OpPq is equal to that of the chain polytope C pPq in [3]. In the present paper we give an explicit description of faces of 2-dimensional simplex of OpPq and C pPq in terms of vertices. We show that triangles in 1-skeleton of OpPq or C pPq are in one-to-one correspondence with faces of 2-dimensional simplex of each polytope. These results are a direct generalizations of [4] (Lemma 4, Lemma 5)
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