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# Compatible Algebras with Straightening Laws on Distributive Lattices

by 1 and 2,*
1
Faculty of Mathematics and Computer Science, University of Bucharest, Academiei 14, Bucharest 010014, Romania
2
Faculty of Mathematics and Computer Science, Ovidius University, Bd. Mamaia 124, Constanta 900527, Romania
*
Author to whom correspondence should be addressed.
Mathematics 2019, 7(8), 671; https://doi.org/10.3390/math7080671
Received: 17 June 2019 / Revised: 24 July 2019 / Accepted: 26 July 2019 / Published: 27 July 2019
(This article belongs to the Special Issue Current Trends on Monomial and Binomial Ideals)

## Abstract

We characterize the finite distributive lattices on which there exists a unique compatible algebra with straightening laws.

## 1. Introduction

Let P be a finite partially ordered set (poset for short) and $I ( P )$ the distributive lattice of the poset ideals of $P .$ A subset $α$ of P is a poset ideal of P if it satisfies the following condition: for every $x ∈ α$ and $y ∈ P$, if $y ≤ x$, then $y ∈ α .$ By a famous theorem of Birkhoff [1], for every finite distributive lattice L, there exists a unique subposet P of L such that $L ≅ I ( P ) .$ The order polytope $O ( P )$ and the chain polytope $C ( P )$ were introduced in [2]. In [3], it was shown that the toric ring $K [ O ( P ) ]$ over a field K is an algebra with straightening laws (ASL in brief) on the distributive lattice $I ( P )$ over the field $K .$ In [4], it was shown that the ring $K [ C ( P ) ]$ associated with the chain polytope shares the same property.
Let $S = K [ x 1 , … , x n , t ]$ be the polynomial ring over a field K and ${ w α } α ∈ I ( P )$ be an arbitrary set of monomials in $x 1 , … , x n$ indexed by $I ( P )$. Let $K [ Ω ] ⊂ S$ be the toric ring generated over K by the set of monomials $Ω = { ω α } α ∈ I ( P )$ where $ω α = w α t$ for all $α ∈ I ( P ) .$ Clearly, $K [ Ω ]$ is a graded algebra if we set $deg ( ω α ) = 1$ for all $α ∈ I ( P ) .$ Let $φ : I ( P ) → K [ Ω ]$ be the injective map defined by $φ ( α ) = ω α$ for all $α ∈ I ( P ) .$ Assume that $K [ Ω ]$ is an ASL on $I ( P )$ over $K .$ According to [4], $K [ Ω ]$ is a compatible ASL if each of its straightening relations is of the form $φ ( α ) φ ( α ′ ) = φ ( β ) φ ( β ′ )$ with $β ⊆ α ∩ α ′$ and $β ′ ⊇ α ∪ α ′ ,$ where $α , α ′$ are incomparable elements in $I ( P ) .$ If $K [ Ω ]$ and $K [ Ω ′ ]$ are compatible ASL on $I ( P )$ over $K ,$ we identify them if they have the same straightening relations. In this case, we write $K [ Ω ] ≡ K [ Ω ′ ] .$
In ([4], Question 5.1), Hibi and Li asked the following questions:
(a)
Given a finite poset P, find all possible compatible algebras with straightening laws on $I ( P )$ over $K .$
(b)
For which posets P, does there exist a unique compatible ASL on $I ( P )$ over $K ?$
In this note, we give a complete answer to question (b). Namely, we prove the following:
Theorem 1.
Let P be a finite poset. Then, the following statements are equivalent:
(i)
There exists a unique compatible ASL on$I ( P )$over$K .$
(ii)
$K [ O ( P ) ] ≡ K [ C ( P ) ] ≡ K [ C ( P * ) ] ,$where$P *$denotes the dual poset of$P .$
(iii)
Each connected component of P is a chain, that is, P is a direct sum of chains.
An answer to question (a) seems to be quite difficult. In ([4], Example 5.2), it was observed that, if one considers the poset $P = { a , b , c , d , e }$ with $a < c < e$ and $b < c < d ,$ then there exist nine compatible ASL structures on $I ( P )$ over $K ,$ while if one considers $P = { a , b , c , d }$ with $a < c$, $b < c , b < d$, then there are three compatible ASL structures on $I ( P )$ over $K ,$ namely, $K [ O ( P ) ] ,$ $K [ C ( P ) ]$, and $K [ C ( P * ) ] .$

## 2. Order Polytopes, Chain Polytopes, and Their Associated Toric Rings

Let $P = { p 1 , … , p n }$ be a finite poset. For the basic terminology regarding posets used in this paper, we refer to [1] and ([5], Chapter 3). The order polytope $O ( P )$ is defined as
$O ( P ) = { ( x 1 , … , x n ) ∈ R n : 0 ≤ x i ≤ 1 , 1 ≤ i ≤ n , and x i ≥ x j if p i ≤ p j in P } .$
In ([2], Corollary 1.3), it was shown that the vertices of $O ( P )$ are $∑ p i ∈ α e i , α ∈ I ( P ) .$ Here, $e i$ denotes the unit coordinate vector in $R n .$ If $α = ∅ ,$ then the corresponding vertex in $O ( P )$ is the origin of $R n .$
The chain polytope $C ( P )$ is defined as
$C ( P ) = { ( x 1 , … , x n ) ∈ R n : x i ≥ 0 , 1 ≤ i ≤ n ,$
$x i 1 + ⋯ + x i r ≤ 1 if p i 1 < ⋯ < p i r is a maximal chain in P } .$
In ([2], Theorem 2.2), it was proved that the vertices of $C ( P )$ are $∑ p i ∈ A e i$, where A is an antichain in $P .$ Recall that an antichain in P is a subset of P such that any two distinct elements in the subset are incomparable. Since every poset ideal is uniquely determined by its antichain of maximal elements, it follows that $O ( P )$ and $C ( P )$ have the same number of vertices. However, as it was observed in [2], $O ( P )$ and $C ( P )$ need not have the same number of i-dimensional faces for $i > 0 .$ Therefore, in general, they are not combinatorial equivalent. Combinatorially, equivalence of order and chain polytopes are studied in [6].

#### The Toric Rings $K [ O ( P ) ]$ and $K [ C ( P ) ]$

To each subset $W ⊂ P$, we attach the squarefree monomial $u W ∈ K [ x 1 , … , x n ] ,$ $u W = ∏ p i ∈ W x i .$ If $W = ∅ ,$ then $u W = 1 .$ The toric ring $K [ O ( P ) ]$, known as the Hibi ring associated with the distributive lattice $I ( P ) ,$ is generated over K by all the monomials $u α t ∈ S$, where $α ∈ I ( P ) .$ The toric ring $K [ C ( P ) ]$ is generated by all the monomials $u A t$ where A is an antichain in $P .$ In addition, as we have already mentioned in the Introduction, both rings are algebras with straightening laws on $I ( P )$ over $K .$
We recall the definition of an ASL as it was introduced in [7]. For a quick introduction to this topic, we refer to [7] and ([8], Chapter XIII). Algebras with straightening laws turned out to be useful tools in studying determinantal rings. Let K be a field, $R = ⨁ i ≥ 0 R i$ with $R 0 = K$ be a graded K-algebra, H a finite poset, and $φ : H → R$ an injective map which maps each $α ∈ H$ to a homogeneous element $φ ( α ) ∈ R$ with $deg φ ( α ) ≥ 1$. A standard monomial in R is a product $φ ( α 1 ) φ ( α 2 ) ⋯ φ ( α k )$ where $α 1 ≤ α 2 ≤ ⋯ ≤ α k$ in $H .$
Definition 1.
The K-algebra R is called an algebra with straightening laws on H over K if the following conditions hold:
(1)
The set of standard monomials is a K–basis of$R ;$
(2)
If$α , β ∈ H$are incomparable and if$φ ( α ) φ ( β ) = ∑ c i φ ( γ i 1 ) … φ ( γ i k i ) ,$where$c i ∈ K \ { 0 }$and$γ i 1 ≤ … ≤ γ i k i ,$is the unique expression of$φ ( α ) φ ( β )$as a linear combination of standard monomials, then$γ i 1 ≤ α , β$for all$i .$
The above relations $φ ( α ) φ ( β ) = ∑ c i φ ( γ i 1 ) … φ ( γ i k i )$ are called the straightening relations of R and they generate all the relations of of $R .$
Let us go back to the toric rings $K [ O ( P ) ]$ and $K [ C ( P ) ] .$
One considers $φ : I ( P ) → K [ O ( P ) ]$ defined by $φ ( α ) = u α t$ for every $α ∈ I ( P ) .$ As it was proved by Hibi in [3], $K [ O ( P ) ]$ is an ASL on $I ( P )$ over K with the straightening relations $φ ( α ) φ ( β ) = φ ( α ∩ β ) φ ( α ∪ β ) ,$ where $α , β$ are incomparable elements in $I ( P ) .$
On the other hand, one defines $ψ : I ( P ) → K [ C ( P ) ]$ by setting $ψ ( α ) = u max α t$ for all $α ∈ I ( P )$ where $max α$ denotes the set of the maximal elements in $α .$ Note that, for every $α ∈ I ( P ) ,$ $max α$ is an antichain in P and each antichain $A ⊂ P$ determines a unique ideal $α ∈ I ( P ) ,$ namely, the poset ideal generated by $A .$ Therefore, $ψ$ is an injective well defined map and by ([4], Theorem 3.1), the ring $K [ C ( P ) ]$ is an ASL on $I ( P )$ over K with the straightening relations
$ψ ( α ) ψ ( β ) = ψ ( α ∗ β ) ψ ( α ∪ β ) ,$
where $α ∗ β$ is the poset ideal of P generated by $max ( α ∩ β ) ∩ ( max α ∪ max β ) .$
We observe that one may also consider $K [ C ( P * ) ]$ as an ASL on $I ( P )$, where $P *$ is the dual poset of $P .$ We may define $δ : I ( P ) → K [ C ( P * ) ]$ by $δ ( α ) = u min α ¯ t$ for $α ∈ I ( P ) ,$ where $min α ¯$ is the set of minimal elements in $α ¯$ and $α ¯$ is the filter $P \ α$ of $P .$ We recall that a filter $γ$ in P (or dual order ideal) is a subset of P with the property that for every $p ∈ γ$ and every $q ∈ P$ with $q ≥ p ,$ we have $q ∈ γ .$ Thus, a filter in P is simply a poset ideal in the dual poset $P * .$ The ring $K [ C ( P * ) ]$ is an ASL on $I ( P )$ over K as well with the straightening relations
$δ ( α ) δ ( β ) = δ ( α ∩ β ) δ ( α ∘ β )$
for incomparable elements $α , β ∈ I ( P ) ,$ where $α ∘ β$ is the poset ideal of P which is the complement in P of the filter generated by $min ( α ¯ ∩ β ¯ ) ∩ ( min α ¯ ∪ min β ¯ ) .$ Let us also observe that all the algebras $K [ O ( P ) ] , K [ C ( P ) ]$, and $K [ C ( P * ) ]$ are compatible algebras with straightening laws.

## 3. Proof of Theorem 1

We clearly have (i) ⇒ (ii). Let us now prove (ii) ⇒ (iii). By hypothesis, the straightening relations of $K [ O ( P ) ] , K [ C ( P ) ] ,$ and $K [ C ( P * ) ]$ coincide. Therefore, we must have
$α ∩ β = α ∗ β and α ¯ ∩ β ¯ = α ∘ β ¯$
for all $α , β$ incomparable elements in $I ( P ) .$ From the second equality in (1), it follows that $α ¯ ∩ β ¯$ is the filter of P generated by $min ( α ¯ ∩ β ¯ ) ∩ ( min α ¯ ∪ min β ¯ ) .$ Assume that there exists two incomparable elements $p , p ′ ∈ P$ such that there exists $q ∈ P$ with $q > p$ and $q > p ′ .$ Consider $α ¯$ the filter generated by p and $β ¯$ the filter generated by $p ′ .$ Then, $min ( α ¯ ∩ β ¯ ) ∩ ( min α ¯ ∪ min β ¯ ) = ∅$, but obviously, $α ¯ ∩ β ¯ ≠ ∅ .$ This shows that, for any two incomparable elements $p , p ′ ∈ P ,$ there is no upper bound for p and $p ′ .$
Similarly, by using the first equality in Equation (1), we derive that, for any two incomparable elements $p , p ′ ∈ P ,$ there is no lower bound for p and $p ′$. This shows that every connected component of the poset P is a chain.
Finally, we prove (iii) ⇒ (i). Let P be a poset such that all its connected components are chains and assume that the cardinality of P is equal to $n .$ Let ${ ω α } α ∈ I ( P )$ be the generators of $K [ Ω ] ⊂ S$ and assume that the straightening relations of $K [ Ω ]$ are $φ ( α ) φ ( α ′ ) = φ ( β ) φ ( β ′ )$ where $β ⊆ α ∩ α ′ ,$ $β ′ ⊇ α ∪ α ′ ,$ and $α , α ′$ are incomparable elements in $I ( P ) .$ We have to show that, for all $α , α ′$ incomparable elements in $I ( P ) ,$ we have $β = α ∩ α ′$ and $β ′ = α ∪ α ′ .$
We proceed by induction on
$k = n − ( rank ( α ∪ α ′ ) − rank ( α ∩ α ′ ) ) .$
Let us recall that, if $γ ∈ I ( P ) ,$ then $rank γ$ denotes the rank of the subposet of $I ( P )$ consisting of all elements $δ ∈ I ( P )$ with $δ ⊆ γ .$
If $k = 0 ,$ that is, $rank ( α ∪ α ′ ) − rank ( α ∩ α ′ ) = n$, then $α ∪ α ′ = P$ and $α ∩ α ′ = ∅ ,$ thus $β = α ∩ α ′$ and $β ′ = α ∪ α ′ .$ Assume that the desired conclusion is true for $rank ( α ∪ α ′ ) − rank ( α ∩ α ′ ) = n − k$ with $k ≥ 0 .$ Let us choose now $α , α ′$ incomparable in $I ( P )$ such that $rank ( α ∪ α ′ ) − rank ( α ∩ α ′ ) = n − k − 1$ and assume that we have a straightening relation $φ ( α ) φ ( α ′ ) = φ ( β ) φ ( β ′ )$ with $β ⊊ α ∩ α ′$ or $β ′ ⊋ α ∪ α ′ .$ By duality, we may reduce to considering $β ′ ⊋ α ∪ α ′ .$ In other words, in $K [ Ω ] ,$ we have
$ω α ω α ′ = ω β ω β ′ , with β ⊆ α ∩ α ′ and β ′ ⊋ α ∪ α ′ .$
As P is a direct sum of chains, we may find $p ∈ max ( α ∪ α ′ )$ and $q ∈ β ′ \ ( α ∪ α ′ )$ such that q covers p in P, that is, $q > p$ and there is no other element $q ′$ in P with $q > q ′ > p .$ Without loss of generality, we may assume that $p ∈ α ′ .$ Let $α 1$ be the poset ideal of P generated by $α ′ ∪ { q } .$ As all the connected components of P are chains, we have $α 1 = α ′ ∪ { q }$ since there are no other elements in P which are smaller than q except those that are on the same chain as p and q, which are in $α ′ .$ Moreover, by the choice of $q ,$ we have
$α 1 ⊆ β ′ and α ∩ α 1 = α ∩ ( α ′ ∪ { q } ) = α ∩ α ′ .$
On the other hand,
$rank ( α ∪ α 1 ) − rank ( α ∩ α 1 ) = rank ( α ∪ α ′ ∪ { q } ) − rank ( α ∩ α ′ )$
$= rank ( α ∪ α ′ ) + 1 − rank ( α ∩ α ′ ) = n − k .$
By the inductive hypothesis, it follows that $φ ( α ) φ ( α 1 ) = φ ( α ∩ α 1 ) φ ( α ∪ α 1 ) ,$ or, equivalently, in $K [ Ω ]$ we have the equality $ω α ω α 1 = ω α ∩ α 1 ω α ∪ α 1 .$ Thus, we have obtained the following equalities in $K [ Ω ] :$
$ω α ω α ′ = ω β ω β ′ and ω α ω α 1 = ω α ∩ α ′ ω α ∪ α 1 .$
This implies that
$ω α ∩ α ′ ω α ′ ω α ∪ α 1 = ω β ω α 1 ω β ′ .$
$α ∩ α ′ ⊂ α ′ ⊂ α ∪ α 1 and β ⊂ α ∩ α ′ = α ∩ α 1 ⊂ α 1 ⊂ β ′ .$
This implies that the monomials in Equation (2) are distinct standard monomials in $K [ Ω ] ,$ which is in contradiction to the condition that the standard monomials form a K-basis in $K [ Ω ] .$ Therefore, our proof is completed.

## Author Contributions

The authors have the same contribution.

## Funding

This research received no external funding.

## Acknowledgments

We would like to thank the anonymous referees for their valuable comments.

## Conflicts of Interest

The authors declare no conflict of interest.

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