Let P be a finite partially ordered set (poset for short) and the distributive lattice of the poset ideals of A subset of P is a poset ideal of P if it satisfies the following condition: for every and , if , then By a famous theorem of Birkhoff , for every finite distributive lattice L, there exists a unique subposet P of L such that The order polytope and the chain polytope were introduced in . In , it was shown that the toric ring over a field K is an algebra with straightening laws (ASL in brief) on the distributive lattice over the field In , it was shown that the ring associated with the chain polytope shares the same property.
Let be the polynomial ring over a field K and be an arbitrary set of monomials in indexed by . Let be the toric ring generated over K by the set of monomials where for all Clearly, is a graded algebra if we set for all Let be the injective map defined by for all Assume that is an ASL on over According to , is a compatible ASL if each of its straightening relations is of the form with and where are incomparable elements in If and are compatible ASL on over we identify them if they have the same straightening relations. In this case, we write
In (, Question 5.1), Hibi and Li asked the following questions:
Given a finite poset P, find all possible compatible algebras with straightening laws on over
For which posets P, does there exist a unique compatible ASL on over
In this note, we give a complete answer to question (b). Namely, we prove the following:
Let P be a finite poset. Then, the following statements are equivalent:
There exists a unique compatible ASL onover
wheredenotes the dual poset of
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 (, Example 5.2), it was observed that, if one considers the poset with and then there exist nine compatible ASL structures on over while if one considers with , , then there are three compatible ASL structures on over namely, , and
2. Order Polytopes, Chain Polytopes, and Their Associated Toric Rings
Let be a finite poset. For the basic terminology regarding posets used in this paper, we refer to  and (, Chapter 3). The order polytope is defined as
In (, Corollary 1.3), it was shown that the vertices of are Here, denotes the unit coordinate vector in If then the corresponding vertex in is the origin of
The chain polytope is defined as
In (, Theorem 2.2), it was proved that the vertices of are , where A is an antichain in 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 and have the same number of vertices. However, as it was observed in , and need not have the same number of i-dimensional faces for Therefore, in general, they are not combinatorial equivalent. Combinatorially, equivalence of order and chain polytopes are studied in .
The Toric Rings and
To each subset , we attach the squarefree monomial If then The toric ring , known as the Hibi ring associated with the distributive lattice is generated over K by all the monomials , where The toric ring is generated by all the monomials where A is an antichain in In addition, as we have already mentioned in the Introduction, both rings are algebras with straightening laws on over
We recall the definition of an ASL as it was introduced in . For a quick introduction to this topic, we refer to  and (, Chapter XIII). Algebras with straightening laws turned out to be useful tools in studying determinantal rings. Let K be a field, with be a graded K-algebra, H a finite poset, and an injective map which maps each to a homogeneous element with . A standard monomial in R is a product where in
The K-algebra R is called an algebra with straightening laws on H over K if the following conditions hold:
The set of standard monomials is a K–basis of
Ifare incomparable and ifwhereandis the unique expression ofas a linear combination of standard monomials, thenfor all
The above relations are called the straightening relations of R and they generate all the relations of of
Let us go back to the toric rings and
One considers defined by for every As it was proved by Hibi in , is an ASL on over K with the straightening relations where are incomparable elements in
On the other hand, one defines by setting for all where denotes the set of the maximal elements in Note that, for every is an antichain in P and each antichain determines a unique ideal namely, the poset ideal generated by Therefore, is an injective well defined map and by (, Theorem 3.1), the ring is an ASL on over K with the straightening relations
where is the poset ideal of P generated by
We observe that one may also consider as an ASL on , where is the dual poset of We may define by for where is the set of minimal elements in and is the filter of We recall that a filter in P (or dual order ideal) is a subset of P with the property that for every and every with we have Thus, a filter in P is simply a poset ideal in the dual poset The ring is an ASL on over K as well with the straightening relations
for incomparable elements where is the poset ideal of P which is the complement in P of the filter generated by Let us also observe that all the algebras , and 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 and coincide. Therefore, we must have
for all incomparable elements in From the second equality in (1), it follows that is the filter of P generated by Assume that there exists two incomparable elements such that there exists with and Consider the filter generated by p and the filter generated by Then, , but obviously, This shows that, for any two incomparable elements there is no upper bound for p and
Similarly, by using the first equality in Equation (1), we derive that, for any two incomparable elements there is no lower bound for p and . 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 Let be the generators of and assume that the straightening relations of are where and are incomparable elements in We have to show that, for all incomparable elements in we have and
We proceed by induction on
Let us recall that, if then denotes the rank of the subposet of consisting of all elements with
If that is, , then and thus and Assume that the desired conclusion is true for with Let us choose now incomparable in such that and assume that we have a straightening relation with or By duality, we may reduce to considering In other words, in we have
As P is a direct sum of chains, we may find and such that q covers p in P, that is, and there is no other element in P with Without loss of generality, we may assume that Let be the poset ideal of P generated by As all the connected components of P are chains, we have 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 Moreover, by the choice of we have
On the other hand,
By the inductive hypothesis, it follows that or, equivalently, in we have the equality Thus, we have obtained the following equalities in
This implies that
In addition, we have:
This implies that the monomials in Equation (2) are distinct standard monomials in which is in contradiction to the condition that the standard monomials form a K-basis in Therefore, our proof is completed.
The authors have the same contribution.
This research received no external funding.
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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