Faces of 2-Dimensional Simplex of Order and Chain Polytopes

: 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.


Introduction
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]. Moreover, in [2], the problem when the order polytope OpPq and the chain polytope C pPq are unimodularly equivalent is solved. It is also 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. In other words, 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).

Definition and Known Results
Let P " tx 1 , . . . , x d u be a finite poset. To each subset W Ă P, we associate ρpWq " ř iPW e i P R d , where e 1 , . . . , e d are the canonical unit coordinate vectors of R d . In particular ρpHq is the origin of R d . A poset ideal of P is a subset I of P such that, for all x i and x j with x i P I and x j ď x i , one has x j P I. An antichain of P is a subset A of P such that x i and x j belonging to A with i ‰ j are incomparable. The empty set H is a poset ideal as well as an antichain of P. We say that x j covers x i if x i ă x j and x i ă x k ă x j for no x k P P. A chain x j 1 ă x j 2 ă¨¨¨ă x j of P is called saturated if x j q covers x j q´1 for 1 ă q ď . A maximal chain is a saturated chain such that x j 1 is a minimal element and x j is a maximal element of the poset. The rank of P is 7pCq´1, where C is a chain with maximum length of P.
The order polytope of P is the convex polytope OpPq Ă R d which consists of those pa 1 , . . . a d q P R d such that 0 ď a i ď 1 for every 1 ď i ď d together with The chain polytope of P is the convex polytope C pPq Ă R d which consists of those pa 1 , . . . , a d q P R d such that a i ě 0 for every 1 ď i ď d together with a i i`a i 2`¨¨¨`a i k ď 1 for every maximal chain x i 1 ă x i 2 ă¨¨¨ă x i k of P.
One has dimOpPq " dimC pPq " d. The vertices of OpPq is those ρpIq for which I is a poset ideal of P ([1] (Corollary1.3)) and the vertices of C pPq is those ρpAq for which A is an antichain of P ([1] (Theorem2.2)). It then follows that the number of vertices of OpPq is equal to that of C pPq. Moreover, the volume of OpPq and that of C pPq are equal to epPq{d!, where epPq is the number of linear extensions of P ([1] (Corollary4.2)). It also follows from [1] that the facets of OpPq are the following: and that the facets of C pPq are the following: In [4] a characterization of edges of OpPq and those of C pPq is obtained. Recall that a subposet Q of finite poset P is said to be connected in P if, for each x and y belonging to Q, there exists a sequence x " x 0 , x 1 , . . . , x s " y with each x i P Q for which x i´1 and x i are comparable in P for each 1 ď i ď s.

1.
Let I and J be poset ideals of P with I ‰ J. Then the convex hull of tρpIq, ρpJqu forms an edge of OpPq if and only if I Ă J and JzI is connected in P.

2.
Let A and B be antichains of P with A ‰ B. Then the convex hull of tρpAq, ρpBqu forms an edge of C pPq if and only if pAzBq Y pBzAq is connected in P.

Faces of 2-Dimensional Simplex
Using Lemma 1, we show the following description of faces of 2-dimensional simplex.
Theorem 1. Let P be a finite poset. Let I, J, and K be pairwise distinct poset ideals of P. Then the convex hull of tρpIq, ρpJq, ρpKqu forms a 2-face of OpPq if and only if I Ă J Ă K and KzI is connected in P.

Proof. ("Only if")
If the convex hull of tρpIq, ρpJq, ρpKqu forms a 2-face of OpPq, then the convex hulls of tρpIq, ρpJqu, tρpJq, ρpKqu, and tρpIq, ρpKqu form edges of OpPq. It then follows from Lemma 1 that I Ă J Ă K and KzI is connected in P.
("If") Suppose that the convex hull of tρpIq, ρpJq, ρpKqu has dimension 1. Then there exists a line passing through the lattice points ρpIq, ρpJq, and ρpKq. Hence ρpIq, ρpJq, and ρpKq cannot be vertices of OpPq. Thus the convex hull of tρpIq, ρpJq, ρpKqu has dimension 2.
Let P " tx 1 , . . . , x d u. If there exists a maximal element x i of P not belonging to I Y J Y K, then the convex hull of tρpIq, ρpJq, ρpKqu lies in the facet x i " 0. If there exists a minimal element x j of P belonging to I X J X K, then the convex hull of tρpIq, ρpJq, ρpKqu lies in the facet x j " 1. Hence, working with induction on dpě 2q, we may assume that I Y J Y K " P and I X J X K " H. Suppose that H " I Ă J Ă K " P and KzI " P is connected. Case 1. 7pJq " 1.
Let J " tx i u and P 1 " Pztx i u. Then P 1 is a connected poset. Let x i 1 , . . . , x i q be the maximal elements of P and A ij " ty P P 1 | y ă x i j u , where 1 ď j ď q. Then we write We then claim that the hyperplane H of R d defined by the equation hpxq " ř d k"1 b k x k " 0 is a supporting hyperplane of OpPq and that H X OpPq coincides with the convex hull of tρpHq, ρpJq, ρpPqu. Clearly hpρpHqq " hpρpPqq " 0 and hpρpJqq " b i " 0. Let I be a poset ideal of P with I ‰ H, I ‰ P and I ‰ J. We have to prove that hpρpIqq ą 0. To simplify the notation, suppose that I X tx i 1 , . . . , x i q u " tx i 1 , . . . , x i r u, where 0 ď r ă q. If r " 0, then hpρpJqq ą 0. Let 1 ď r ă q, I 1 " Iztx i u, and K " Ť r j"1 pA i j Y tx i j uq. Then I 1 and K are poset ideals of P and hpρpKqq ď hpρpI 1 qq " hpρpIqq. We claim hpρpKqq ą 0. One has hpρpKqq ě 0. Moreover, hpρpKqq " 0 if and only if no z P K belongs to Let PzJ " tx i u and P 1 " Pztx i u. Then P 1 is a connected poset. Thus we can show the existence of a supporting hyperplane of OpPq which contains the convex hull of tρpHq, ρpJq, ρpPqu by the same argument in Case 1.
To simplify the notation, suppose that J " tx 1 , . . . , x u. Then PzJ " tx `1 , . . . , x d u. Since J and PzJ are subposets of P, these posets are connected. Let x i 1 , . . . , x i q be the maximal elements of J and x i q`1 , . . . , x i q`r the maximal elements of PzJ. Then we write . . , i q , i q`1 , . . . , i q`r ú 7pA ij q if k P ti 1 , . . . , i q , i q`1 , . . . , i q`r u .
We then claim that the hyperplane H of R d defined by the equation hpxq " ř d k"1 b k x k " 0 is a supporting hyperplane of OpPq and H X OpPq coincides with the convex hull of tρpHq, ρpJq, ρpPqu. Clearly hpρpHqq " hpρpJqq " hpρpPzJqq " 0, then hpρpPqq " hpρpJqq`hpρpPzJqq " 0. Let I be a poset ideal of P with I ‰ H, I ‰ P and I ‰ J. What we must prove is hpρpIqq ą 0.
If I Ă J, then I is a poset ideal of J. To simplify the notation, suppose that I X tx i 1 , . . . , x i q u " tx i 1 , . . . , x i s u , where 0 ď s ă q. If s " 0, then hpρpIqq ą 0. Let 1 ď s ă q, K " Ť s j"1 pA i j Y tx i j uq. Then K is a poset ideal of J and hpρpKqq ď hpρpIqq. Thus we can show hpρpKqq ą 0 by the same argument in Case 1 (Replace r with s and P 1 with J).
If J Ă I, then IzJ is a poset ideal of PzJ. To simplify the notation, suppose that pIzJq X tx i q`1 , . . . , x i q`r u " tx i q`1 , . . . , x i q`t u , where 0 ď t ă r. If t " 0, then hpρpIqq " hpρpJqq`hpρpIzJqq " hpρpIzJqq ą 0. Let 1 ď t ă r, K " Ť q`t j"q`1 pA i j Y tx i j uq. Then K is a poset ideal of PzJ and hpρpKqq ď hpρpIzJqq " hpρpIqq. Thus we can show hpρpKqq ą 0 by the same argument in Case 1 (Replace r with q`t , q with q`r and P 1 with PzJ). Consequently, hpρpIqq ą 0, as desired.
Let A B denote the symmetric difference of the sets A and B, that is A B " pAzBq Y pBzAq.

Proof. ("Only if")
If the convex hull of tρpAq, ρpBq, ρpCqu forms a 2-face of C pPq, then the convex hulls of tρpAq, ρpBqu, tρpBq, ρpCqu, and tρpAq, ρpCqu form edges of C pPq. It then follows from Lemma 1 that A B, B C and C A are connected in P.
("If") Suppose that the convex hull of tρpAq, ρpBq, ρpCqu has dimension 1. Then there exists a line passing through the lattice points ρpAq, ρpBq, and ρpCq. Hence ρpAq, ρpBq, and ρpCq cannot be vertices of C pPq. Thus the convex hull of tρpAq, ρpBq, ρpCqu has dimension 2.
Let P " tx 1 , . . . , then the convex hull of tρpAq, ρpBq, ρpCqu lies in the facet x i " 0. Furthermore, if A Y B Y C " P and A X B X C ‰ H, then x j P A X B X C is isolated in P and x j itself is a maximal chain of P. Thus the convex hull of tρpAq, ρpBq, ρpCqu lies in the facet x j " 1. Hence, working with induction on dpě 2q, we may assume that A Y B Y C " P and A X B X C " H. As stated in the proof of [3] ([Theorem 2.1]), if A B is connected in P, then A and B satisfy either (i) B Ă A or (ii) y ă x whenever x P A and y P B are comparable. Hence, we consider the following three cases: (a) If B Ă A, then A B " AzB is connected in P, and thus 7pAzBq " 1. Let AzB " tx k u.
In this case, we may assume z ă x if x P A and z P C are comparable. Furthermore, P has rank 1.
(b) If B Ć A and B X A ‰ H, then we may assume y ă x if x P A and y P B are comparable. If C Ă B with C X A X B " H, then as stated in (a), C A cannot be connected. Since C Ć B, we may assume z ă y if y P B and z P C are comparable. If C X B ‰ H, then C X A " H and P has rank 1 or 2.
Similarly, if C X B " H, then C X A " H and P has rank 2.
(c) Let B Ć A and B X A " H. We may assume that if x P A and y P B are comparable, then y ă x. If C Ă B, then we regard this case as equivalent to (a). Let C Ć B. We may assume z ă y if y P B and z P C are comparable. Moreover, if C X B ‰ H, then we regard this case as equivalent to (b). If C X B " H, then C X A " H and P has rank 2.
Consequently, there are five cases as regards antichains for C pPq. Case 1. B Ă A, C X A " H, and C X B " H.
For each x i P B we write b i for the number of elements z P C with z ă x i . For each x j P C we write c j for the number of elements y P B with x j ă y. Let a k " 0 for AzB " tx k u. Clearly where q is the number of pairs py, zq with y P B, z P C and z ă y. Let hpxq " x j PC c j x j`ak x k and let H be the hyperplane of R d defined by hpxq " q. Then hpρpAqq " hpρpBqq " hpρpCqq " q. We claim that, for any antichain D of P with D ‰ A, D ‰ B, and D ‰ C, one has hpρpDqq ă q. Let D " B 1 Y C 1 or D " tx k u Y C 1 with B 1 Ĺ B and C 1 Ĺ C. Suppose D " B 1 Y C 1 . Since B C is connected and since D is an antichain of P, it follows that ř H, and P has rank 1. We define four numbers as follows: γ " 7pty P BzC | x ă y , x P CzBuq.
Since P has rank 1, B Ă A Y C " P. It follows that A " pAzBq Y pBzCq, C " pBzAq Y pCzBq. Then where q 1 is the number of pairs px, yq with x P AzB, y P BzA and y ă x, q 2 is the number of pairs py, zq with y P BzC, z P CzB and z ă y, and q " q 1`q2 . Let and H the hyperplane of R d defined by hpxq " q. Then hpρpAqq " hpρpBqq " hpρpCqq " q. We claim that, for any antichain D of P with D ‰ A, D ‰ B and D ‰ C, one has hpρpDqq ă q. Let D " D 1 Y D 2 with D 1 is an antichain of A B and D 2 is an antichain of B C. Since A B, B C are connected, it follows that hpρpD 1 qq ă q 1 and hpρpD 2 qq ă q 2 . Thus hpρpDqq " hpρpD 1 qq`hpρpD 2 qq ă q 1`q2 " q. Case 3. B Ć A, B X A ‰ H, C Ć B, C X B ‰ H, C X A " H, and P has rank 2.
For each x i P P we write cpiq for the number of maximal chains, which contain x i . Let q be the number of maximal chains in P. Since each x i P A is maximal element and each x k P C is minimal element, Let hpxq " ř x i PP cpiqx i and H the hyperplane of R d defined by hpxq " q. Then hpρpAqq " hpρpBqq " hpρpCqq " q. We claim that, for any antichain D of P with D ‰ A, D ‰ B and D ‰ C, one has hpρpDqq ă q. D " A 1 Y B 1 Y C 1 with A 1 Ă AzB, B 1 Ĺ B, and C 1 Ĺ CzB. Now, we define two subsets of B: Let ř x i PA cpiq " q 1 , ř x j PB 1 cpjq " q 2 , ř x k PC 1 cpkq " q 3 , ř x j PB 2 cpjq " q 1 1 , and ř x j PB 3 cpjq " q 1 3 . Since A B, B C are connected, it follows that q 1 ă q 1 1 and q 3 ă q 1 3 . Hence Thus hpρpDqq ă q. Case 4. B Ć A, B X A ‰ H, C X B " H, and C X A " H. Since P has rank 2, we can show hpρpDqq ă q by the same argument in Case 3 (Suppose C X B " H).
Case 5. B Ć A, B X A " H, C X B " H and C X A " H. Since P has rank 2, we can show hpρpDqq ă q by the same argument in Case 3 (Suppose B X A " C X B " H).
In conclusion, each H is a supporting hyperplane of C pPq and H X C pPq coincides with the convex hull of tρpAq, ρpBq, ρpCqu, as desired. Corollary 1. Triangles in 1-skeleton of OpPq or C pPq are in one-to-one correspondence with faces of 2-dimensional simplex of each polytope.