Join-meet binomial algebras of distributive lattices

JOIN-MEET BINOMIAL ALGEBRAS OF DISTRIBUTIVE LA TTICES BARBARA BETTI AND T AKA YUKI HIBI Abstract. W e in v estigate the defining ideal of the algebra ov er a field generated by the join-meet binomials coming from a finite distributiv e lattice. In the frame of algebras with straigh tening la ws, the problem when the defining ideal is generated b y quadrics is studied. Introduction Let L b e a finite lattice on n elements x 1 , . . . , x n and S = K [ x 1 , . . . , x n ] b e the p olynomial ring in n v ariables o v er a field K . W e are interested in the quadratic binomials f ij = x i x j − ( x i ∨ x j )( x i ∧ x j ) , 1 ≤ i < j ≤ n, called join-me et binomials of L . Clearly , f ij is not zero if and only if x i and x j are incomparable in L . The ideal I L = ( f ij : 1 ≤ i < j ≤ n ) and its quotien t ring S/I L w as in tro duced in [ Hib87 ]. In particular, by virtue of the classical results of Birkhoff [ HH11 , Theorem 9.1.7] and Dedekind [ Sta12 , Chapter 3, Exercise 30], it is shown that I L is a prime ideal if and only if L is a distributive lattice and that, when L is a distributiv e lattice, S/I L is normal and Cohen–Macaula y . Both the ideal I L and the quotient ring S/I L of a distributive lattice hav e b een widely inv estigated thereafter. In this paper we adopt a differen t p erspective and focus on the finitely generated K -algebra R K ( L ) = K [ f ij : 1 ≤ i < j ≤ n, f ij  = 0] ⊂ S of a distributive lattice L . W e do not assume any restriction on the field K . W e refer to R K ( L ) as the join-me et algebr a of L . This class of algebras is of relev an t interest as it con tains imp ortan t examples in comm utative algebra and algebraic geometry suc h as the homogeneous coordinate rings of the Grassmannian Gr K (2 , m ), for ev ery m ≥ 4. The defining ideal of K [Gr K (2 , m )] is generated b y the w ell-known Pl ¨ uc ker quadrics. In general, the defining ideal of R K ( L ) con tains quadrics which are independent of Pl¨ uc ker quadrics. F or example, the join-meet algebra R K ( B 3 ) of the b oolean lattice B 3 of rank 3 (Figure 1 ) has defining ideal generated b y the following t w o quadrics: f 3 , 6 f 4 , 5 + f 2 , 7 f 4 , 6 − f 3 , 5 f 4 , 6 − f 2 , 6 f 4 , 7 = f 2 , 7 f 3 , 5 − f 2 , 7 f 4 , 6 + f 2 , 6 f 4 , 7 − f 2 , 3 f 5 , 7 = 0 . A relev ant result on this class of algebras is giv en in [ HMT26 ], where the authors c harac- terize distributiv e lattices L for whic h the set of generators B L = { f ij : 1 ≤ i < j ≤ n, f ij  = 0 } is a Khov anskii/Sagbi basis with resp ect to a compatible term order. 2020 Mathematics Subje ct Classific ation. 06A11, 13H10, 13P10, 14M15. Key wor ds and phr ases. distributive lattice, binomial, Pl¨ uc k er quadric, p olynomial ring, algebra with straigh tening la ws. The research for this pap er was initiated while the second author stay ed at the Max Planck Institute for Mathematics in the Sciences, Leipzig, August 15 – Septem b er 8, 2025. 1 2 BARBARA BETTI AND T AKA YUKI HIBI 1 2 5 8 7 3 4 6 Figure 1. Bo olean lattice B 3 of rank 3. A final goal of our pro ject is to classify those distributiv e lattices L for whic h the defining ideal of R K ( L ) is generated by quadrics. W e w ork within the framew ork of algebras with straigh tening la ws (ASL) [ Eis80 ]. As a first step in this direction, w e classify distributiv e lat- tices L whose join-meet algebra R K ( L ) is a p olynomial ring (Theorem 3.2 ) and show that the defining ideal of R K ( L ) of a thin distributive lattice is generated by quadrics (Theorem 3.3 ). In the App endix, assuming the Pl ¨ uck er relation [ Bru+22 , p. 73] is known, we give a short pro of of the classical fact (Ho dge [ Ho d43 ] and Doubilet–Rota–Stein [ DRS74 ]) that the homo- geneous coordinate ring K [ x i y j − x j y i : 1 ≤ i < j ≤ n ] of the Grassmann v ariet y Gr K ( d, n ) parameterizing d -dimensional subspaces of K n is an ASL on a distributive lattice o ver K . 1. P ar tiall y ordered sets, distributive la ttices and Gr ¨ obner bases A partially ordered set is called a p oset . A chain of a finite p oset P is a totally ordered subset of P . The length of a c hain C is | C | − 1, where | C | is the cardinality of C . The r ank of P is the biggest length of chains of P . Let rank( P ) denote the rank of P . A finite p oset is called pur e if all of its maximal c hains hav e the same length. A finite lattic e is a finite p oset whose an y tw o elements a, b ha ve a unique greatest low er b ound a ∧ b and a unique least upp er b ound a ∨ b , called, resp ectiv ely , the me et and the join of a and b . A finite lattice is distributive if meet and join op erations distribute o ver each other. A subset I of a finite p oset P is called a p oset ide al of P if it is do wnw ard closed, that is, for ev ery a ∈ I and b ∈ P satisfying b ≤ a in P , w e hav e b ∈ I . In particular, ∅ and P are p oset ideals of P . Let J ( P ) denote the finite p oset consisting of all p oset ideals of P , ordered b y inclusion. Then J ( P ) is a distributive lattice of rank | P | . Let L b e a finite distributiv e lattice and 0 L (resp. 1 L ) b e its unique minimal (resp. maximal) elemen t. W e say that x ∈ L is an ap ex of L if x is comparable to all a ∈ L . In particular, b oth 0 L and 1 L are apexes of L. A finite distributive lattice L is called simple if there is no ap ex of L except for 0 L and 1 L . An elemen t 0 L  = a ∈ L is called join-irr e ducible if a = b ∨ c , then either a = b or a = c . Let P L b e the subposet of L consisting of all join-irreducible elemen ts of L . Birkhoff ’s fundamen tal structure theorem for finite distributive lattices [ HH11 , Theorem 9.1.7] guarantees that L and J ( P L ) are isomorphic as finite lattices. W e sa y that L is planar if among an y three join-irreducible elements a, b, c ∈ L , t wo of them are comparable in L . The r ank of a ∈ L is the maximal length of chains of the form a i 0 < a i 1 < · · · < a i r = a . JOIN-MEET BINOMIAL ALGEBRAS OF DISTRIBUTIVE LA TTICES 3 Let rank L ( a ) denote the rank of a ∈ L . Let d = rank( L ) = rank L (1 L ) and write ρ L ( i ) for the n umber of elements a ∈ L with rank L ( a ) = i , where 0 ≤ i ≤ d . In particular, ρ L (0) = ρ L ( d ) = 1. Let θ ( L ) = max { ρ L ( i ) : 1 ≤ i < d } . A finite simple distributive lattice L is called thin [ HH23 , p. 3] (or a gener alize d snake p oset [ Bel+22 ]) if θ ( L ) = 2, i.e., ρ L ( i ) = 2 for 1 ≤ i < d . It follo ws that a thin distributiv e lattice is planar. Let n > 0 b e an integer and let D n b e the set of all divisors of n , ordered by divisibility . W e call D n the divisor lattic e of n . Every divisor lattice is distributive. A divisor lattice D n is called Bo ole an if n is squarefree. A divisor lattice D n is thin if and only if n = pq s , where p and q are distinct primes and s ≥ 1. The follo wing lemma on Gr¨ obner bases will be used in the next section. W e refer the reader to [ HH11 , Chapter 2] for fundamental materials on Gr¨ obner bases. Lemma 1.1. L et f 1 , . . . , f s b e homo gene ous p olynomials of S = K [ x 1 , . . . , x n ] of the same de gr e e > 0 and let < b e a monomial or der on S . Consider the subrings K [ f 1 , . . . , f s ] and K [in < ( f 1 ) , . . . , in < ( f s )] of S . L et A = K [ y 1 , . . . , y s ] b e the p olynomial ring in s variables over K . We define the surje ctive ring homomorphisms ϕ : A → K [ f 1 , . . . , f s ] and ψ : A → K [in < ( f 1 ) , . . . , in < ( f s )] by setting ϕ ( y i ) = f i and ψ ( y i ) = in < ( f i ) . L et I = Ker( ϕ ) denote the defining ide al of K [ f 1 , . . . , f s ] and J = Ker( ψ ) the defining ide al of K [in < ( f 1 ) , . . . , in < ( f s )] . L et < ♯ b e a monomial or der on A and G ′ = { g 1 , . . . , g t } a Gr¨ obner b asis of J with r esp e ct to < ♯ . Supp ose that in < ♯ ( g j ) ∈ in < ♯ ( I ) for e ach 1 ≤ j ≤ t and cho ose h j ∈ I with in < ♯ ( g j ) = in < ♯ ( h j ) . Then G := { h 1 , . . . , h t } is a Gr¨ obner b asis of I with r esp e ct to < ♯ . Pr o of. Let H denote the set of those monomials u ∈ A with u ∈ (in < ♯ ( h 1 ) , . . . , in < ♯ ( h t )). It suffices to prov e that { ϕ ( u ) : u ∈ H} is a linearly indep enden t set. Since (in < ♯ ( h 1 ) , . . . , in < ♯ ( h t )) = (in < ♯ ( g 1 ) , . . . , in < ♯ ( g t )) = in < ♯ ( J ) , it follo ws that { ψ ( u ) : u ∈ H } is linearly independent. Let u 1 , . . . , u δ b e monomials, where u ξ  = u ξ ′ if ξ  = ξ ′ , b elonging to H with δ > 1. Thus ψ ( u ξ )  = ψ ( u ξ ′ ) if ξ  = ξ ′ . Supp ose that δ X ξ =1 c ξ ϕ ( u ξ ) = δ X ξ =1 c ξ f α (1) ξ 1 · · · f α ( s ) ξ s = 0 , 0  = c ξ ∈ K. W e observe that in < ( f α (1) ξ 1 · · · f α ( s ) ξ s ) = (in < ( f 1 )) α (1) ξ · · · (in < ( f s )) α ( s ) ξ = ψ ( u ξ ) and, since ψ ( u ξ )  = ψ ( u ξ ′ ) whenev er ξ  = ξ ′ , there exists a unique 1 ≤ ξ 0 ≤ δ for which 0 = in <   δ X ξ =1 c ξ ϕ ( u ξ )   = c ξ 0 (in < ( f 1 )) α (1) ξ 0 · · · (in < ( f s )) α ( s ) ξ 0 = c ξ 0 ψ ( u ξ 0 ) . This is a contradiction, hence G is a Gr¨ obner basis of I with resp ect to < ♯ . □ R emark 1.2 . In Lemma 1.1 , if in < ( f 1 ) , . . . , in < ( f s ) are algebraically indep enden t, then f 1 , . . . , f s are algebraically indep endent. 4 BARBARA BETTI AND T AKA YUKI HIBI 2. Algebras with straightening la ws Let R = L ∞ n =0 R n b e a no etherian graded algebra o v er K . Let P b e a finite p oset and supp ose that an injection ϕ : P  → S ∞ n =1 R n for whic h the K -algebra R is generated by ϕ ( P ) ov er K is given. A standar d monomial is a homogeneous element of R of the form ϕ ( γ 1 ) ϕ ( γ 2 ) · · · ϕ ( γ n ), where γ 1 ≤ γ 2 ≤ · · · ≤ γ n in P . W e call R an algebr as with str aightening laws [ Eis80 ] on P o ver K if the follo wing conditions are satisfied: (ASL -1) The set of standard monomials is a basis of R ov er K ; (ASL -2) If α and β in P are incomparable and ϕ ( α ) ϕ ( β ) = X i r i ϕ ( γ i 1 ) ϕ ( γ i 2 ) · · · , 0  = r i ∈ K, γ i 1 ≤ γ i 2 ≤ · · · (2.1) is the unique expression of ϕ ( α ) ϕ ( β ) ∈ R as linear com bination of distinct standard monomials guaran teed by (ASL -1), then γ i 1 ≤ α, β for ev ery i . The right-hand side of the relation in (ASL -2) is allow ed to b e the empt y sum (= 0). W e abbreviate an algebra with straightening laws as ASL. The relations in (ASL -2) are called the str aightening r elations for R . Let A = K [ x α : α ∈ P ] denote the p olynomial ring in | P | v ariables o v er K and define the surjection π : A → R b y setting π ( x α ) = ϕ ( α ). The defining ideal I R of R = L ∞ n =0 R n is the k ernel Ker( π ) of π . If α and β in P are incomparable, then we introduce the p olynomial of A f α,β := x α x β − X i r i x γ i 1 x γ i 2 · · · arising from (ASL-2) and belonging to I R . Let G R denote the set of those f α,β for whic h α and β are incomparable in P . Let < rev denote the reverse lexicographic order on A induced b y an ordering of the v ariables for which x α < rev x β if α < β in P . It follo ws that in < rev ( f α,β ) = x α x β and G R is a Gr¨ obner basis of I R with resp ect to < rev . F urthermore, dim R = rank( P ) + 1. In the definition of ASL, if we require only (ASL -2), then we call R a we akly ASL on P o ver K . It would be of interest to find a criterion for a w eakly ASL on P o ver K to b e an ASL on P o ver K . One possible and standard w a y is to compute Hilb ert functions. On the other hand, by virtue of Lemma 1.1 , w e hav e the follo wing result. Lemma 2.1. L et f 1 , . . . , f s b e homo gene ous p olynomials of S = K [ x 1 , . . . , x n ] of the same de gr e e > 0 and let < b e a monomial or der on S . Consider the subrings K [ f 1 , . . . , f s ] and K [in < ( f 1 ) , . . . , in < ( f s )] of S . L et P = { p 1 , . . . , p s } b e a finite p oset and define the inje ction π : P → K [ f 1 , . . . , f s ] and π ′ : P → K [in < ( f 1 ) , . . . , in < ( f s )] by setting π ( p i ) = f i and π ′ ( p i ) = in < ( f i ) . Supp ose that K [in < ( f 1 ) , . . . , in < ( f s )] is an ASL on P over K and that K [ f 1 , . . . , f s ] is a we akly ASL on P over K . Then K [ f 1 , . . . , f s ] is an ASL on P over K . Pr o of. Let G ′ = { f ′ α,β : α , β uncomparable } be the Gr¨ obner basis of Ker( π ′ ) defined via the straigh tening relations of K [in < ( f 1 ) , . . . , in < ( f s )]. Then the p olynomials f α,β , defined after the straightening relations of K [ f 1 , . . . , f s ], share the initial terms with the p olynomials of G ′ , hence they form a Gr¨ obner basis for Ker( π ) by Lemma 1.1 . Since the defining ideal of K [ f 1 , . . . , f s ] has a Gr¨ obner basis consisting of straightening relations, the standard monomials are linearly indep endent and K [ f 1 , . . . , f s ] is ASL on P ov er K . □ Example 2.2. Let P n = { ( i, j ) : 1 ≤ i, j ≤ n } denote the finite p oset whose partial order is defined b y setting ( i, j ) ≤ ( i ′ , j ′ ) if i ≤ i ′ and j ≤ j ′ . Let S = K [ x 1 , . . . , x n , y 1 , . . . , y n ] denote JOIN-MEET BINOMIAL ALGEBRAS OF DISTRIBUTIVE LA TTICES 5 ˆ 0 x 1 x 2 x 3 x 4 x 5 x 6 x 7 x 8 x 9 ˆ 1 y 9 y 8 y 7 y 6 y 5 y 4 y 3 y 2 y 1 Figure 2. A thin distributive lattice. the polynomial ring in 2 n v ariables o ver K and R n = K [ x i y j : 1 ≤ i, j ≤ n ]. After defining the injection π : P n → R n b y setting π (( i, j )) = x i y j , it follo ws from the discussion done in [ Hib87 , pp. 98–99] that R n is an ASL on P n o ver K . Let L = { 0 L , x 1 , . . . , x n , y 1 , . . . , y n , 1 L } b e a thin distributive lattice with n + 1 = rank( L ), where 0 L < x 1 < · · · < x n < 1 L and 0 L < y 1 < · · · < y n < 1 L (Figure 2 ). Note that, once x 1 and y 1 are fixed, the assignmen t of x i and y i for 2 ≤ i ≤ n is automatically determined. Let S = K [ x 1 , . . . , x n , y 1 , . . . , y n ] denote the polynomial ring in 2 n v ariables o v er K . Recall that w e asso ciate x i and y j whic h are incomparable in L with the quadratic binomial f ij = x i y j − ( x i ∧ y j )( x i ∨ y j ) of S . W e fix a monomial order < on S for which in < ( f ij ) = x i y j for all f ij . W e introduce the subalgebra R K ( L ) of S which is generated b y all binomials f ij . Let Q L b e the sublattice of P n of Example 2.2 consisting of the pairs ( i, j ) with f ij ∈ R K ( L ) (Figure 3 ). Lemma 2.3 b elo w is a simple consequence of Birkhoff ’s fundamental structure theorem for finite distributiv e lattices [ HH11 , Theorem 9.1.7]. Lemma 2.3. The p artial or der on { x i − 1 , y i − 1 , x n , y n } is either x i − 1 < x i , y i − 1 < y i , x i − 1 < y i or x i − 1 < x i , y i − 1 < y i , y i − 1 < x i . Lemma 2.4. The subp oset Q L of a thin distributive lattic e L of r ank n + 1 is a chain of the p oset P n of Example 2.2 if and only if the divisor lattic e D 2 · 3 3 is not a sublattic e of L . 6 BARBARA BETTI AND T AKA YUKI HIBI x 1 x 2 x 3 x 4 x 5 x 6 x 7 x 8 x 9 y 1 y 2 y 3 y 4 y 5 y 6 y 7 y 8 y 9 Figure 3. The sublattice Q L of the thin distributive lattice L of Figure 2 Pr o of. Supp ose that D 2 · 3 3 = { 2 p · 3 q : p ∈ { 0 , 1 } , q ∈ { 0 , 1 , 2 , 3 }} is a sublattice of L . Then (2 , 3 2 ) < (2 · 3 , 3 2 ) < (2 · 3 , 3 3 ) and (2 , 3 2 ) < (2 , 3 3 ) < (2 · 3 , 3 3 ). Thus Q L cannot b e a c hain. No w, supp ose that the divisor lattice D 2 · 3 3 is not a sublattice of L . Let, say , x i − 2 < x i − 1 , x i − 2 < y i − 1 , y i − 2 < y i − 1 . W orking with induction on n enables us to assume that the sublattice Q L ′ with L ′ = L \ { x i , y i } is a chain of P n whose maximal elemen t is ( x i − 1 , y i − 1 ). Since y i − 1 < x i , y i − 1 < y i , x i − 1 < x i , it follows that Q L = Q L ′ ∪ { ( x i − 1 , y i ) , ( x i , y i ) } is a chain of P n , as desired. □ Example 2.5. Let L = D 2 · 3 n − 1 = { ˆ 0 , x 1 , . . . , x n − 1 , y 1 , . . . , y n − 1 , ˆ 1 } with 2 = y 1 , 3 = x 1 as ab o ve. If ˆ 0 = x 0 and ˆ 1 = y n . Let ( , i ) and ( k , j ) are incomparable in Q L , sa y , i < j ≤ k <  , then the classical Pl ¨ uck er relation yields ( y i x ℓ − y ℓ +1 x i − 1 )( y j x k − y k +1 x j − 1 ) = ( y i x k − y k x i − 1 )( y j x ℓ − y ℓ x j − 1 ) − ( y i x j − 1 − y j x i − 1 )( y k +1 x ℓ − y ℓ +1 x k ) , In other words, we hav e a quadratic relation among the binomial generators of R K ( L ): f ℓ,i f k,j = f k,i f ℓ,j − f j − 1 ,i f ℓ,k +1 . Since ( k , i ) < ( , i ) , ( j − 1 , i ) < ( , i ) , ( k , i ) < ( k, j ) , ( j − 1 , i ) < ( k , j ), it follows that R K ( L ) is a w eakly ASL on Q L o ver K . Theorem 2.6. The join-me et algebr a R K ( L ) of a thin distributive lattic e L is a we akly ASL on Q n over K . Pr o of. Assume x 1 < x 2 , x 1 < y 2 , y 1 < y 2 and let i 0 b e the smallest in teger i for which x i − 2 < x i − 1 , x i − 2 < y i − 1 , y i − 2 < y i − 1 and y i − 1 < x i , y i − 1 < y i , x i − 1 < x i . Let L ( − ) denote the interv al [ ˆ 0 , x i 0 ] of L and L (+) the interv al [ x i 0 − 2 , ˆ 1] of L . Then ( x i 0 − 1 , y i 0 − 1 ) is the unique JOIN-MEET BINOMIAL ALGEBRAS OF DISTRIBUTIVE LA TTICES 7 Figure 4. Divisor lattice D 2 2 · 3 2 . maximal element of Q L ( − ) as well as the unique minimal element of Q L (+) . F urthermore, each α ∈ Q L ( − ) and eac h β ∈ Q L (+) are comparable in L . Thus ( x i 0 − 1 , y i 0 − 1 ) is an ap ex of Q L . No w, Example 2.5 says that R K ( L ( − ) ) is a weakly ASL on Q L ( − ) o ver K and, in addition, w orking with induction on n , it follo ws that R K ( L (+) ) is a weakly ASL on Q L (+) o ver K . Hence R K ( L ) is a weakly ASL on Q L o ver K , as required. □ 3. Rela tions of join-meet binomials Let L b e a finite distributive lattice on n elements x 1 , . . . , x n and S = K [ x 1 , . . . , x n ] b e the p olynomial ring in n v ariables ov er a field K . Recall that the join-meet binomials of L are f ij = x i x j − ( x i ∨ x j )( x i ∧ x j ) , 1 ≤ i < j ≤ n. Let R K ( L ) b e the join-meet algebra R k ( L ) = k [ f ij : 1 ≤ i < j ≤ n, f ij  = 0] ⊂ S of L . Let A = K [ y i,j : 1 ≤ i < j ≤ n, f ij  = 0] denote the p olynomial ring o v er K and define the surjectiv e ring homomorphism π : A → R K ( L ) b y setting π ( y ij ) = f ij . The defining ideal I R K ( L ) of R K ( L ) is the kernel of π . It is natural to ask for whic h distributiv e lattices L the join-meet algebra R K ( L ) is isomorphic to a p olynomial ring, that is, the join-meet binomials are algebraically indep endent. Example 3.1. The divisor lattice D 2 2 · 3 2 (Figure 4 ) has 9 binomial generators. These are the 2-minors of a 3 × 3 matrix. It follows from [ Bru+22 , Theorem 6.4.7 (c)] that the join-meet algebra R K ( D 2 , 2 ) is a p olynomial ring. Theorem 3.2. L et L b e a distributive lattic e. Then R K ( L ) is a p olynomial ring if and only if L is planar and the divisor lattic e D 2 · 3 3 is not a sublattic e of L . Pr o of. If L is non-planar, then the b o olean lattice B 3 of rank 3 is a sublattice of L . As w as seen in Introduction, R K ( B 3 ) cannot b e a p olynomial ring. Th us R K ( L ) cannot b e a p olynomial ring. Let L b e planar. If the divisor lattice D 2 · 3 3 is a sublattice of L , then a Pl ¨ uc k er relation app ears in the defining ideal I R K ( L ) of R K ( L ), as shown in Example 2.5 . Supp ose that L is planar and that D 2 · 3 3 is not a sublattice of L . If L is thin, then R K ( L ) is a p olynomial ring (Lemma 2.4 ). If L is not thin and D 2 · 3 3 is not a sublattice of L , then L = D 2 2 · 3 2 and the desired result is Example 3.1 . □ 8 BARBARA BETTI AND T AKA YUKI HIBI As mentioned in Introduction, a final goal of our pro ject is to classify those distributive lattices L for whic h the defining ideal I R K ( L ) of R K ( L ) is generated b y quadrics. As first step to ward this purp ose, we show that the defining ideal I R K ( L ) of a thin distributiv e lattice L is generated b y quadrics. Recall that a finite simple distributiv e lattice L is thin if θ ( L ) = 2. Theorem 3.3. Supp ose that a finite simple distributive lattic e L is thin. Then the defin- ing ide al I R K ( L ) of R K ( L ) is gener ate d by quadrics and has a quadr atic Gr¨ obner b asis. In p articular, R K ( L ) is a Koszul algebr a. F urthermor e, R K ( L ) is Gor enstein. Pr o of. Lemma 2.1 together with Theorem 2.6 guarantees that the defining ideal I R K ( L ) of R K ( L ) is generated by quadrics that form a Gr¨ obner basis. In particular, b y a result of F r¨ ob erg, R K ( L ) is Koszul [ F r ¨ o99 ]. F urthermore, since the p oset Q n on which R K ( L ) is ASL is pure, it follows from the Gorenstein criterion that R K ( L ) is Gorenstein [ Hib87 , p. 105]. □ Example 3.4. Let L and L ′ b e the nonplanar distributiv e lattices of Figure 5 and Figure 6 , resp ectively . They b oth satisfy θ ( L ) = θ ( L ′ ) = 3. Ho wev er, the defining ideal I R K ( L ) of R K ( L ) is generated b y quadrics, but the defining ideal I R K ( L ′ ) of R K ( L ′ ) cannot b e generated b y quadrics. Figure 5. Nonplanar distributive lattice with θ ( L ) = 3 Lemma 3. 5. L et L b e a finite simple planar distributive lattic e and supp ose that θ ( L ) = r . Then the divisor lattic e D 2 r − 1 · 3 r − 1 is an interval of L . Pr o of. Let P L b e the poset of all join-irreducible elemen ts of L . One has L = J ( P L ). Since L is planar, it follows from Dilworth’s theorem [ Sta12 , Chapter 3, Exercise 77] that P L can b e decomp osed into the disjoin t union of c hains C and C ′ of P L . Let C : x 0 < · · · < x t and C ′ : y 0 < · · · < y s . If a ∈ L is equal to I ∈ J ( P L ) and if x p (resp. y q ) is the maximal elemen t of I ∩ C (resp. I ∩ C ′ ), then w e emplo y the notation a = ( x p , y q ), where p = − 1 if I ∩ C = ∅ . Now, supp ose that ρ L ( i ) = r ≥ 2. Let a 1 , . . . , a r denote the elements b elonging to { a ∈ L : rank L ( a ) = i } . Let a j = ( x p j , y q j ) , 1 ≤ j ≤ r , where p j + q j = i − 2. Let p 1 < · · · < p r and q 1 > · · · > q r . Since a 1 = ( x p 1 , y q 1 ) and a r = ( x p r , y q r ) are p oset ideals of P L , it follo ws that eac h x ∈ C with x p 1 < x ≤ x p r and eac h y ∈ C ′ with y q r < y ≤ y q 1 are incomparable in P L . In particular, b oth p 1 , . . . , p r and q r , . . . , q 1 are successiv e integers. F urthermore, I ∗ = { x 0 , x 1 , . . . , x p 1 , y 0 , y 1 , . . . , y q r } , I ∗ = { x 0 , x 1 , . . . , x p r , y 0 , y 1 , . . . , y q 1 } are poset ideals of P L . Let a ∗ = ( x p 1 , y q r ) ∈ L and a ∗ = ( x p r , y q 1 ) ∈ L . It then follo ws that the in terv al [ a ∗ , a ∗ ] of L is the divisor lattice D 2 r − 1 · 3 r − 1 . □ JOIN-MEET BINOMIAL ALGEBRAS OF DISTRIBUTIVE LA TTICES 9 Figure 6. Planar distributive lattice with θ ( L ′ ) = 3 Lemma 3.6. L et L b e a finite simple planar distributive lattic e and supp ose that ther e is i 0 with ρ L ( i 0 ) = ρ L ( i 0 + 1) = 3 . Then the divisor lattic e D 2 2 · 3 3 is an interval of L . Pr o of. W e follow the pro of of Lemma 3.5 . Let a = ( x p − 1 , y q +1 ) , b = ( x p , y q ) , c = ( x p +1 , y q − 1 ) b e the elements of L with rank L ( a ) = rank L ( b ) = rank L ( c ) = i 0 . Then a ′ = ( x p , y q +1 ) , c ′ = ( x p +1 , y q ) are elemen ts of L with rank L ( a ′ ) = rank L ( c ′ ) = i 0 + 1. Since ρ L ( i 0 + 1) = 3, it follows that either a ′′ = ( x p +2 , y q − 1 ) or c ′′ = ( x p − 1 , y q +2 ) m ust b elong L . Say , a ′′ = ( x p +2 , y q − 1 ) ∈ L . Hence x p < x p +1 < x p +2 and y q < y q +1 are the disjoint union of c hains. Th us the in terv al [ f , g ] of L , where f = ( x p − 1 , y q − 1 ) and g = ( x p +2 , y q +1 ), is the divisor lattice D 2 2 · 3 3 . □ Lemma 3.7. L et L b e a finite distributive lattic e and let L ′ b e an interval of L . If the defining ide al I R K ( L ) of R K ( L ) is gener ate d by quadrics, then I R K ( L ′ ) is gener ate d by quadrics. Pr o of. W e first recall that, since L ′ is an interv al of L , it follows that α, β ∈ L b elong to L ′ if and only if α ∧ β and α ∨ β b elong to L ′ . W e claim R K ( L ′ ) ⊂ R K ( L ) is an algebra retract [ OHH00 , p. 747]. Let S = K [ x i : x i ∈ L ] and S ′ = K [ x j : x j ∈ L ′ ] be p olynomial rings o ver K . W e consider the natural epimorphism p : S → S ′ , fixing x i if it b elongs to L ′ and mapping x i to 0 otherwise. If α, β are in L ′ , then p ( f ij ) = f ij , otherwise, one among x α ∨ β and x α ∧ β is not in L ′ and p ( f ij ) = 0. Th us the image of R K ( L ) under p is R K ( L ′ ) and the restriction of p to R K ( L ′ ) is the identit y . It follows that ε = p |R K ( L ′ ) is a retraction map for R K ( L ′ ) ⊂ R K ( L ). The desired result follows from comparing the graded Betti num b ers of the defining ideals of R K ( L ′ ) and R K ( L ) [ OHH00 , Corollary 2.5]. □ Theorem 3.8. L et L b e a finite simple planar distributive lattic e and supp ose that one of the fol lowing c onditions is satisfie d: (i) θ ( L ) > 3 ; (ii) θ ( L ) = 3 and ther e is i 0 with ρ L ( i 0 ) = ρ L ( i 0 + 1) = 3 . Then the defining ide al I R K ( L ) of R K ( L ) c annot b e gener ate d by quadrics. Pr o of. Lemmas 3.6 and 3.5 guaran tee that an in terv al of L is the divisor lattice D 2 2 · 3 3 . 10 BARBARA BETTI AND T AKA YUKI HIBI In [ BCV13 ] the authors pro v ed that the ideal I R K ( D 2 2 · 3 3 ) has minimally cubic relations, whic h, together with the Pl ¨ uck er quadrics, form a system of generators [ Hua+21 ]. Then it follows from Lemma 3.7 that I R K ( L ) cannot b e generated b y quadrics. □ W e close the presen t pap er with the following conjecture. Conjecture 3.9. L et L b e a simple planar distributive lattic e. Then the defining ide al I R K ( L ) of R K ( L ) is gener ate d by quadrics if and only if at most one interval is D 2 2 · 3 2 . Appendix W e giv e a short proof of the classical result that the homogeneous co ordinate ring of the Grassmannian Gr K ( d, n ) is an ASL on a distributiv e lattice ov er K . Let 2 ≤ d ≤ n b e integers. Let L denote the set of those symbols [ i 1 · · · i d ] with 1 ≤ i 1 < · · · < i d ≤ n . W e introduce the partial order on L by setting [ i 1 · · · i d ] ≤ [ i ′ 1 · · · i ′ d ] if i 1 ≤ i ′ 1 , . . . , i d ≤ i ′ d . It then follows that L is a distributiv e lattice and [ i 1 · · · i d ] ∈ L is join-irreducible if and only if [ i 1 · · · i d ] is one of the following: • [ a, a + 1 , . . . , a + ( d − 1)] , 1 < a ≤ n − d + 1; • [1 , 2 , . . . , a, a + b, a + b + 1 , · · · , b + d − 1] , 1 < a < d, 1 < b ≤ n − d + 1. Let X = { x ij } 1 ≤ i ≤ d, 1 ≤ j ≤ n denote the d × n matrix of v ariables and [ i 1 , . . . , i d ], where 1 ≤ i 1 < · · · < i d ≤ n , the determinant of the d × d submatrix of X consisting of columns i 1 , . . . , i d . Let S = K [ x ij : 1 ≤ i ≤ d, 1 ≤ j ≤ n ] denote the p olynomial ring in dn v ariables o ver a field K . Let A denote the subring of S generated b y those determinan ts [ i 1 , . . . , i d ] with 1 ≤ i 1 < · · · < i d ≤ n . No w, considering L to be a subset of A in an ob vious w ay . Pl ¨ uck er relations guaran tee that A is a weakly ASL on L ov er K . Since there is a monomial order < on S for which in < ([ i 1 , . . . , i d ]) = x 1 ,i 1 x 2 ,i 2 · · · x d,i d [ Bru+22 , p. 80], b y virtue of Lemma 2.1 , in order to show that A is an ASL on L ov er K , it suffices to pro v e the following Lemma. Lemma 3.10. Define the inje ction ψ : L → S by setting ψ ([ i 1 , . . . , i d ]) = Q d j =1 x j,i j . Then the toric ring K [ ψ ( ξ ) : ξ ∈ L ] is an ASL on L over K . Pr o of. Let P denote the set of join-irreducible elements of L . Then P can b e decomp osed in to the disjoint union of c hains P 1 , P 2 , . . . , P d , where P 1 = { [2 , 3 , . . . , d + 1] , [3 , 4 , . . . , d + 2] , . . . , [ n − d + 1 , n − d + 2 , . . . , n ] } , P 2 = { [1 , 3 , 4 , . . . , d + 1] , [1 , 4 , 5 , . . . , d + 2] , . . . , [1 , n − d + 2 , n − d + 3 , . . . , n ] } , P 3 = { [1 , 2 , 4 , 5 , . . . , d + 2] , [1 , 2 , 5 , 6 , . . . , d + 3] , . . . , [1 , 2 , n − d + 3 , . . . , n ] } , . . . P d = { [1 , 2 , 3 , . . . , d − 1 , d + 1] , [1 , 2 , 3 , . . . , d − 1 , d + 2] , . . . , [1 , 2 , 3 , . . . , d − 1 , n ] } . W e introduce the sym b ols x ij , 1 ≤ i ≤ d, 1 ≤ j ≤ n and express eac h P i in the form P i : x i,i +1 < x i,i +2 < · · · < x i,n − d + i . F or example, in Figure 7 , one has [1345] = x 2 , 3 and [1256] = x 3 , 5 . F urthermore, we add a unique minimal element ˆ 0 = x 1 , 1 x 2 , 2 · · · x d,d to P . W e define the injection ϕ : L → S by setting ϕ ([ i 1 , . . . , i d ]) = f 1 f 2 · · · f d , where f j = x j,j x j,j +1 x j,j +2 · · · x j,i j . REFERENCES 11 [4567] [3456] [2345] [1567] [1456] [1345] [1267] [1256] [1245] [1237] [1236] [1235] Figure 7. The p oset P with d = 4 and n = 7 Since x j,j + q = [1 , 2 , . . . , j − 1 , j + q , j + q + 1 , . . . ] ≤ [ i 1 , . . . , i d ] if and only if j + q ≤ i j , it follo ws that the toric ring K [ ϕ ( p ) : p ∈ L ] coincides with R K ( L ) introduced in [ Hib87 ]. In particular, K [ ϕ ( ξ )] : ξ ∈ L ] is an ASL on L ov er K . On the other hand, defining the injection ψ : L → S b y setting ψ ([ i 1 , . . . , i d ]) = Q d j =1 x j,i j , one clearly has K [ ϕ ( ξ ) : ξ ∈ L ] = K [ ψ ( ξ ) : ξ ∈ L ]. □ References [BCV13] Winfried Bruns, Aldo Conca, and Matteo V arbaro. “Relations betw een the minors of a generic matrix”. In: A dvanc es in Mathematics 244 (2013), pp. 171–206. 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[HH11] J ¨ urgen Herzog and T ak ayuki Hibi. Monomial Ide als . V ol. 260. Graduate T exts in Mathematics. Springer, 2011. 12 REFERENCES [HH23] J ¨ urgen Herzog and T ak a yuki Hibi. “Finite distributiv e lattices, p oly omino es and ideals of K¨ onig type”. In: International Journal of Mathematics 34 (2023). [Hib87] T ak a yuki Hibi. “Distributive lattices, affine semigroup rings and algebras with straigh tening laws”. In: Commutative Algebr a and Combinatorics . Ed. by Masay oshi Nagata and Hideyuki Matsum ura. V ol. 11. Adv anced Studies in Pure Mathemat- ics. Amsterdam: North-Holland, 1987, pp. 93–109. [HMT26] Akihiro Higashitani, Ko ji Matsushita, and Koichiro T ani. “Khov anskii bases of subalgebras arising from finite distributive lattices”. In: Journal of A lgebr a 693 (2026), pp. 805–823. [Ho d43] W. V. D. Ho dge. “Some Enumerativ e Results in the Theory of F orms”. In: Math- ematic al Pr o c e e dings of the Cambridge Philosophic al So ciety 39.1 (1943), pp. 22– 30. [Hua+21] Hang Huang et al. “Relations betw een the 2 × 2 minors of a generic matrix”. In: A dvanc es in Mathematics 386 (2021). [OHH00] Hidefumi Ohsugi, J ¨ urgen Herzog, and T ak ayuki Hibi. “Combinatorial pure sub- rings”. In: Osaka Journal of Mathematics 37 (2000), pp. 745–757. [Sta12] Ric hard Stanley . Enumer ative Combinatorics, V olume I, Se c ond Ed. V ol. 49. Cam- bridge Studies in Adv anced Mathematics. Cam bridge Universit y Press, 2012. Authors’ addresses: (Barbara Betti) Otto-von-Guericke-University Magdebur g, Institut f ¨ ur Algebra und Ge- ometrie, Universit ¨ atspla tz 2, Magdeburg, Germany. Email addr ess : barbara.betti@ovgu.de (T aka yuki Hibi) Dep ar tment of Pure and Applied Ma thema tics, Gradua te School of Inf orma- tion Science and Technology, Osaka University, Suit a, Osaka 565–0871, Jap an. Email addr ess : hibi@math.sci.osaka-u.ac.jp