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Open AccessFeature PaperArticle

Faces of 2-Dimensional Simplex of Order and Chain Polytopes

Department of Pure and Applied Mathematics, Graduate School of Information Science and Technology, Osaka University, Suita, Osaka 565-0871, Japan
Mathematics 2019, 7(9), 851; https://doi.org/10.3390/math7090851
Received: 11 August 2019 / Revised: 9 September 2019 / Accepted: 10 September 2019 / Published: 14 September 2019
(This article belongs to the Special Issue Current Trends on Monomial and Binomial Ideals)

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.
Keywords: order polytope; chain polytope; partially ordered set order polytope; chain polytope; partially ordered set

1. Introduction

The combinatorial structure of the order polytope O ( P ) and the chain polytope C ( P ) of a finite poset (partially ordered set) P is explicitly discussed in [1]. Moreover, in [2], the problem when the order polytope O ( P ) and the chain polytope C ( P ) are unimodularly equivalent is solved. It is also proved that the number of edges of the order polytope O ( P ) is equal to that of the chain polytope C ( P ) in [3]. In the present paper we give an explicit description of faces of 2-dimensional simplex of O ( P ) and C ( P ) in terms of vertices. In other words, we show that triangles in 1-skeleton of O ( P ) or C ( P ) 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).

2. Definition and Known Results

Let P = { x 1 , , x d } be a finite poset. To each subset W P , we associate ρ ( W ) = i W e i R d , where e 1 , , e d are the canonical unit coordinate vectors of R d . In particular ρ ( ) 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 I and x j x i , one has x j 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 is a poset ideal as well as an antichain of P. We say that x j covers x i if x i < x j and x i < x k < x j for no x k P . A chain x j 1 < x j 2 < < x j of P is called saturated if x j q covers x j q 1 for 1 < q . 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 ( C ) 1 , where C is a chain with maximum length of P.
The order polytope of P is the convex polytope O ( P ) R d which consists of those ( a 1 , a d ) R d such that 0 a i 1 for every 1 i d together with
a i a j
if x i x j in P.
The chain polytope of P is the convex polytope C ( P ) R d which consists of those ( a 1 , , a d ) R d such that a i 0 for every 1 i d together with
a i i + a i 2 + + a i k 1
for every maximal chain x i 1 < x i 2 < < x i k of P.
One has dim O ( P ) = dim C ( P ) = d . The vertices of O ( P ) is those ρ ( I ) for which I is a poset ideal of P ([1] (Corollary1.3)) and the vertices of C ( P ) is those ρ ( A ) for which A is an antichain of P ([1] (Theorem2.2)). It then follows that the number of vertices of O ( P ) is equal to that of C ( P ) . Moreover, the volume of O ( P ) and that of C ( P ) are equal to e ( P ) / d ! , where e ( P ) is the number of linear extensions of P ([1] (Corollary4.2)). It also follows from [1] that the facets of O ( P ) are the following:
  • x i = 0 , where x i P is maximal;
  • x j = 1 , where x j P is minimal;
  • x i = x j , where x j covers x i ,
and that the facets of C ( P ) are the following:
  • x i = 0 , for all x i P ;
  • x i 1 + + x i k = 1 , where x i 1 < < x i k is a maximal chain of P.
In [4] a characterization of edges of O ( P ) and those of C ( P ) 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 Q for which x i 1 and x i are comparable in P for each 1 i s .
Lemma 1
([4] (Lemma 4, Lemma 5)) Let P be a finite poset.
1.
Let I and J be poset ideals of P with I J . Then the convex hull of { ρ ( I ) , ρ ( J ) } forms an edge of O ( P ) if and only if I J and J I is connected in P.
2.
Let A and B be antichains of P with A B . Then the convex hull of { ρ ( A ) , ρ ( B ) } forms an edge of C ( P ) if and only if ( A B ) ( B A ) is connected in P.

3. 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 { ρ ( I ) , ρ ( J ) , ρ ( K ) } forms a 2-face of O ( P ) if and only if I J K and K I is connected in P.
Proof. 
(“Only if”) If the convex hull of { ρ ( I ) , ρ ( J ) , ρ ( K ) } forms a 2-face of O ( P ) , then the convex hulls of { ρ ( I ) , ρ ( J ) } , { ρ ( J ) , ρ ( K ) } , and { ρ ( I ) , ρ ( K ) } form edges of O ( P ) . It then follows from Lemma 1 that I J K and K I is connected in P.
(“If”) Suppose that the convex hull of { ρ ( I ) , ρ ( J ) , ρ ( K ) } has dimension 1. Then there exists a line passing through the lattice points ρ ( I ) , ρ ( J ) , and ρ ( K ) . Hence ρ ( I ) , ρ ( J ) , and ρ ( K ) cannot be vertices of O ( P ) . Thus the convex hull of { ρ ( I ) , ρ ( J ) , ρ ( K ) } has dimension 2.
Let P = { x 1 , , x d } . If there exists a maximal element x i of P not belonging to I J K , then the convex hull of { ρ ( I ) , ρ ( J ) , ρ ( K ) } lies in the facet x i = 0 . If there exists a minimal element x j of P belonging to I J K , then the convex hull of { ρ ( I ) , ρ ( J ) , ρ ( K ) } lies in the facet x j = 1 . Hence, working with induction on d ( 2 ) , we may assume that I J K = P and I J K = . Suppose that = I J K = P and K I = P is connected.
Case 1. ( J ) = 1 .
Let J = { x i } and P = P { x i } . Then P is a connected poset. Let x i 1 , , x i q be the maximal elements of P and A i j = { y P | y < x i j } , where 1 j q . Then we write
b k = ( { i j | x k A i j } ) if k { i 1 , , i q , i } 0 if k = i ( A i j ) if k { i 1 , , i q } .
We then claim that the hyperplane H of R d defined by the equation h ( x ) = k = 1 d b k x k = 0 is a supporting hyperplane of O ( P ) and that H O ( P ) coincides with the convex hull of { ρ ( ) , ρ ( J ) , ρ ( P ) } . Clearly h ( ρ ( ) ) = h ( ρ ( P ) ) = 0 and h ( ρ ( J ) ) = b i = 0 . Let I be a poset ideal of P with I , I P and I J . We have to prove that h ( ρ ( I ) ) > 0 . To simplify the notation, suppose that I { x i 1 , , x i q } = { x i 1 , , x i r } , where 0 r < q . If r = 0 , then h ( ρ ( J ) ) > 0 . Let 1 r < q , I = I { x i } , and K = j = 1 r ( A i j { x i j } ) . Then I and K are poset ideals of P and h ( ρ ( K ) ) h ( ρ ( I ) ) = h ( ρ ( I ) ) . We claim h ( ρ ( K ) ) > 0 . One has h ( ρ ( K ) ) 0 . Moreover, h ( ρ ( K ) ) = 0 if and only if no z K belongs to A i r + 1 A i q . Now, since P is connected, it follows that there exists z K with z A i r + 1 A i q . Hence h ( ρ ( K ) ) > 0 . Thus h ( ρ ( I ) ) > 0 .
Case 2. ( J ) = d 1 .
Let P J = { x i } and P = P { x i } . Then P is a connected poset. Thus we can show the existence of a supporting hyperplane of O ( P ) which contains the convex hull of { ρ ( ) , ρ ( J ) , ρ ( P ) } by the same argument in Case 1.
Case 3. 2 ( J ) d 2 .
To simplify the notation, suppose that J = { x 1 , , x } . Then P J = { x + 1 , , x d } . Since J and P J 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 P J . Then we write
A i j = { y J | y < x i j } if 1 j q { y P J | y < x i j } if q + 1 j r
and
b k = ( { i j | x i A i j } ) if k { i 1 , , i q , i q + 1 , , i q + r } ( A i j ) if k { i 1 , , i q , i q + 1 , , i q + r } .
We then claim that the hyperplane H of R d defined by the equation h ( x ) = k = 1 d b k x k = 0 is a supporting hyperplane of O ( P ) and H O ( P ) coincides with the convex hull of { ρ ( ) , ρ ( J ) , ρ ( P ) } . Clearly h ( ρ ( ) ) = h ( ρ ( J ) ) = h ( ρ ( P J ) ) = 0 , then h ( ρ ( P ) ) = h ( ρ ( J ) ) + h ( ρ ( P J ) ) = 0 . Let I be a poset ideal of P with I , I P and I J . What we must prove is h ( ρ ( I ) ) > 0 .
If I J , then I is a poset ideal of J. To simplify the notation, suppose that I { x i 1 , , x i q } = { x i 1 , , x i s } , where 0 s < q . If s = 0 , then h ( ρ ( I ) ) > 0 . Let 1 s < q , K = j = 1 s ( A i j { x i j } ) . Then K is a poset ideal of J and h ( ρ ( K ) ) h ( ρ ( I ) ) . Thus we can show h ( ρ ( K ) ) > 0 by the same argument in Case 1 (Replace r with s and P with J).
If J I , then I J is a poset ideal of P J . To simplify the notation, suppose that ( I J ) { x i q + 1 , , x i q + r } = { x i q + 1 , , x i q + t } , where 0 t < r . If t = 0 , then h ( ρ ( I ) ) = h ( ρ ( J ) ) + h ( ρ ( I J ) ) = h ( ρ ( I J ) ) > 0 . Let 1 t < r , K = j = q + 1 q + t ( A i j { x i j } ) . Then K is a poset ideal of P J and h ( ρ ( K ) ) h ( ρ ( I J ) ) = h ( ρ ( I ) ) . Thus we can show h ( ρ ( K ) ) > 0 by the same argument in Case 1 (Replace r with q + t , q with q + r and P with P J ). Consequently, h ( ρ ( I ) ) > 0 , as desired. □
Let A B denote the symmetric difference of the sets A and B, that is A B = ( A B ) ( B A ) .
Theorem 2.
Let P be a finite poset. Let A, B, and C be pairwise distinct antichains of P. Then the convex hull of { ρ ( A ) , ρ ( B ) , ρ ( C ) } forms a 2-face of C ( P ) if and only if A B , B C and C A are connected in P.
Proof. 
(“Only if”) If the convex hull of { ρ ( A ) , ρ ( B ) , ρ ( C ) } forms a 2-face of C ( P ) , then the convex hulls of { ρ ( A ) , ρ ( B ) } , { ρ ( B ) , ρ ( C ) } , and { ρ ( A ) , ρ ( C ) } form edges of C ( P ) . 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 { ρ ( A ) , ρ ( B ) , ρ ( C ) } has dimension 1. Then there exists a line passing through the lattice points ρ ( A ) , ρ ( B ) , and ρ ( C ) . Hence ρ ( A ) , ρ ( B ) , and ρ ( C ) cannot be vertices of C ( P ) . Thus the convex hull of { ρ ( A ) , ρ ( B ) , ρ ( C ) } has dimension 2.
Let P = { x 1 , , x d } . If A B C P and x i A B C , then the convex hull of { ρ ( A ) , ρ ( B ) , ρ ( C ) } lies in the facet x i = 0 . Furthermore, if A B C = P and A B C , then x j A B C is isolated in P and x j itself is a maximal chain of P. Thus the convex hull of { ρ ( A ) , ρ ( B ) , ρ ( C ) } lies in the facet x j = 1 . Hence, working with induction on d ( 2 ) , we may assume that A B C = P and A B C = . 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 A or (ii) y < x whenever x A and y B are comparable. Hence, we consider the following three cases:
(a) If B A , then A B = A B is connected in P, and thus ( A B ) = 1 . Let A B = { x k } . If C A , then C A = { x k } , since A B C = C B = . Namely x k is isolated in P. Hence B C = B C = A B C = P cannot be connected. Thus C A = . In this case, we may assume z < x if x A and z C are comparable. Furthermore, P has rank 1.
(b) If B A and B A , then we may assume y < x if x A and y B are comparable. If C B with C A B = , then as stated in (a), C A cannot be connected. Since C B , we may assume z < y if y B and z C are comparable. If C B , then C A = and P has rank 1 or 2. Similarly, if C B = , then C A = and P has rank 2.
(c) Let B A and B A = . We may assume that if x A and y B are comparable, then y < x . If C B , then we regard this case as equivalent to (a). Let C B . We may assume z < y if y B and z C are comparable. Moreover, if C B , then we regard this case as equivalent to (b). If C B = , then C A = and P has rank 2.
Consequently, there are five cases as regards antichains for C ( P ) .
Case 1. B A , C A = , and C B = .
For each x i B we write b i for the number of elements z C with z < x i . For each x j C we write c j for the number of elements y B with x j < y . Let a k = 0 for A B = { x k } . Clearly x i B b i = x j C c j = q , where q is the number of pairs ( y , z ) with y B , z C and z < y . Let h ( x ) = x i B b i x i + x j C c j x j + a k x k and let H be the hyperplane of R d defined by h ( x ) = q . Then h ( ρ ( A ) ) = h ( ρ ( B ) ) = h ( ρ ( C ) ) = q . We claim that, for any antichain D of P with D A , D B , and D C , one has h ( ρ ( D ) ) < q . Let D = B 1 C 1 or D = { x k } C 1 with B 1 B and C 1 C . Suppose D = B 1 C 1 . Since B C is connected and since D is an antichain of P, it follows that x i B 1 b i + x j C 1 c j < q . Thus h ( ρ ( D ) ) < q . Suppose that D = { x k } C 1 . It follows that x j C 1 c j + a k = x j C 1 c j < x j C c j = q . Thus h ( ρ ( D ) ) < q .
Case 2. B A , B A , C B , C B , C A = , and P has rank 1.
We define four numbers as follows:
α i = ( { y B A | y < x i , x i A B } ) ; γ j = ( { x A B | x j < x , x j B A } ) ; α k = ( { z C B | z < x k , x k B C } ) ; γ = ( { y B C | x < y , x C B } ) .
Since P has rank 1, B A C = P . It follows that A = ( A B ) ( B C ) , C = ( B A ) ( C B ) . Then
x s A α s = x i A B α i + x k B C α k = q ; x j B A γ j + x k B C α k = q ; x u C γ u = x j B A γ j + x C B γ = q ,
where q 1 is the number of pairs ( x , y ) with x A B , y B A and y < x , q 2 is the number of pairs ( y , z ) with y B C , z C B and z < y , and q = q 1 + q 2 . Let
h ( x ) = x s A α s x s + x u C γ u x u = x i A B α i x i + x j B A γ j x j + x k B C α k x k + x C B γ x
and H the hyperplane of R d defined by h ( x ) = q . Then h ( ρ ( A ) ) = h ( ρ ( B ) ) = h ( ρ ( C ) ) = q . We claim that, for any antichain D of P with D A , D B and D C , one has h ( ρ ( D ) ) < q . Let D = D 1 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 h ( ρ ( D 1 ) ) < q 1 and h ( ρ ( D 2 ) ) < q 2 . Thus h ( ρ ( D ) ) = h ( ρ ( D 1 ) ) + h ( ρ ( D 2 ) ) < q 1 + q 2 = q .
Case 3. B A , B A , C B , C B , C A = , and P has rank 2.
For each x i P we write c ( i ) for the number of maximal chains, which contain x i . Let q be the number of maximal chains in P. Since each x i A is maximal element and each x k C is minimal element, x i A c ( i ) = x k C c ( k ) = q . Then
x j B c ( j ) = x s B A c ( s ) + x t B C c ( t ) + x u B ( A C ) c ( u ) = x s B A c ( s ) + x t B C c ( t ) + x v A B c ( v ) x t B C c ( t ) = x i A c ( i ) = q .
Let h ( x ) = x i P c ( i ) x i and H the hyperplane of R d defined by h ( x ) = q . Then h ( ρ ( A ) ) = h ( ρ ( B ) ) = h ( ρ ( C ) ) = q . We claim that, for any antichain D of P with D A , D B and D C , one has h ( ρ ( D ) ) < q . D = A 1 B 1 C 1 with A 1 A B , B 1 B , and C 1 C B . Now, we define two subsets of B:
B 2 = { x j B | x j < x i , x i A 1 } ; B 3 = { x j B | x k < x j , x k C 1 } .
Then B 1 B 2 = B 1 B 3 = B 2 B 3 = and B 1 B 2 B 3 B 3 . Let x i A c ( i ) = q 1 , x j B 1 c ( j ) = q 2 , x k C 1 c ( k ) = q 3 , x j B 2 c ( j ) = q 1 , and x j B 3 c ( j ) = q 3 . Since A B , B C are connected, it follows that q 1 < q 1 and q 3 < q 3 . Hence
h ( ρ ( D ) ) = x i A 1 c ( i ) + x j B 1 c ( j ) + x k C 1 c ( k ) = q 1 + q 2 + q 3 < q 1 + q 2 + q 3 = x j B 2 c ( j ) + x j B 1 c ( j ) + x j B 3 c ( j ) x j B c ( j ) = q .
Thus h ( ρ ( D ) ) < q .
Case 4. B A , B A , C B = , and C A = .
Since P has rank 2, we can show h ( ρ ( D ) ) < q by the same argument in Case 3 (Suppose C B = ).
Case 5. B A , B A = , C B = and C A = .
Since P has rank 2, we can show h ( ρ ( D ) ) < q by the same argument in Case 3 (Suppose B A = C B = ).
In conclusion, each H is a supporting hyperplane of C ( P ) and H C ( P ) coincides with the convex hull of { ρ ( A ) , ρ ( B ) , ρ ( C ) } , as desired. □
Corollary 1.
Triangles in 1-skeleton of O ( P ) or C ( P ) are in one-to-one correspondence with faces of 2-dimensional simplex of each polytope.

Funding

This research received no external funding.

Conflicts of Interest

The author declares no conflict of interest.

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