Let L be a join-distributive lattice with length n and width(Ji L) \leq k.
There are two ways to describe L by k-1 permutations acting on an n-element
set: a combinatorial way given by P.H. Edelman and R.E. Jamison in 1985 and a
recent lattice theoretical way of the second author. We prove that these two
approaches are equivalent. Also, we characterize join-distributive lattices by
trajectories.Comment: 8 pages, 1 figur

Adaricheva, Kira

Czédli, Gábor

English

arXiv

Let L be a join-distributive lattice with length n and width (Ji L)   k.
There are two ways to describe L by k − 1 permutations acting on an n-element set:
a combinatorial way given by P.H. Edelman and R. E. Jamison in 1985 and a recent
lattice theoretical way of the second author. We prove that these two approaches are
equivalent. Also, we characterize join-distributive lattices by trajectorie

Cz´edli, G´abor

Nazarbayev University Repository

arXiv:1210.3376v1  [math.RA]  11 Oct 2012Notes on the description of join-distributive lattices bypermutationsKira Adaricheva and Ga´bor Cze´dliAbstract. Let L be a join-distributive lattice with length n and width (JiL) ≤ k.There are two ways to describe L by k − 1 permutations acting on an n-element set:a combinatorial way given by P.H. Edelman and R. E. Jamison in 1985 and a recentlattice theoretical way of the second author. We prove that these two approaches areequivalent. Also, we characterize join-distributive lattices by trajectories.Introduction. For x 6= 1 in a finite lattice L, let x∗ denote the join of up-per covers of x. A finite lattice L is join-distributive if the interval [x, x∗] isdistributive for all x ∈ L \ {1}. For other definitions, see K. Adaricheva [2],K. Adaricheva, V.A. Gorbunov and V.I. Tumanov [3], and N. Caspard andB. Monjardet [6], see G. Cze´dli [7, Proposition 2.1 and Remark 2.2] for a re-cent survey, and see (3) before the proof of Corollary 6 later for a particularlyuseful variant. The study of (the duals of) join-distributive lattices goes backto R. P. Dilworth [11], 1940. There were a lot of discoveries and rediscoveriesof these lattices and equivalent combinatorial structures; see [3], [7], B. Mon-jardet [17], and M. Stern [18] for surveys. Note that join-distributivity impliessemimodularity; the origin of this result is the combination of M. Ward [19](see also R. P. Dilworth [11, page 771], where [19] is cited) and S. P. Avann [5](see also P.H. Edelman [13, Theorem 1.1(E,H)], when [5] is recalled).The join-width of L, denoted by width (JiL), is the largest k such thatthere is a k-element antichain of join-irreducible elements of L. As usual, Snstands for the set of permutations acting on the set {1, . . . , n}. There aretwo known ways to describe a join-distributive lattice with join-width k andlength n by k−1 permutations; our goal is to enlighten their connection. Thisconnection exemplifies that Lattice Theory can be applied in Combinatoricsand vice versa. We also give a new characterization of join-distributive lattices.Two constructions. For n ∈ N = {1, 2, . . .} and k ∈ {2, 3, . . .}, let ~σ =〈σ2, . . . , σk〉 ∈ Sk−1n . For convenience, σ1 ∈ Sn will denote the identity per-mutation. In the powerset join-semilattice 〈P ({1, . . . , n});∪〉, consider thesubsemilattice LEJ(~σ) generated by{{σi(1), . . . , σi(j)} : i ∈ {1, . . . , k}, j ∈ {0, . . . , n}}. (1)2010 Mathematics Subject Classification: Primary 06C10; secondary 05E99.Key words and phrases: Join-distributive lattice, semimodular lattice, diamond-free lat-tice, trajectory, permutation.This research was supported by the NFSR of Hungary (OTKA), grant numbers K77432and K83219, and by TA´MOP-4.2.1/B-09/1/KONV-2010-0005.2 K. Adaricheva and G. Cze´dli Algebra univers.Figure 1. An example of LEJ(~σ) and LC(~π)Since it contains ∅, LEJ(~σ) is a lattice, the Edelman-Jamison lattice deter-mined by ~σ. Its definition above is a straightforward translation from thecombinatorial language of P.H. Edelman and R.E. Jamison [14, Theorem 5.2]to Lattice Theory. Actually, the original version in [14] describes a decompo-sition of convex geometries.To present an example, let σ2 =(1 2 3 43 2 4 1)and σ3 =(1 2 3 44 2 1 3).Then ~σ = 〈σ2, σ3〉 ∈ S24 , and LEJ(~σ) is depicted in Figure 1. In the label of anelement in LEJ(~σ), only the part before the comma is relevant; to save space,subsets are denoted by listing their elements without commas. For example,134,111 in the figure stands for the subset {1, 3, 4} of {1, 2, 3, 4}. The chaindefined in (1), apart from its top {1, 2, 3, 4} and bottom ∅, corresponds to theblack-filled small squares for i = 1, the light grey-filled pentagons for i = 2,and the dark grey-filled circles for i = 3. Note that LEJ(~σ) consists of allsubsets of {1, 2, 3, 4} but {2}.Next, we recall a related construction from G. Cze´dli [7]. Given ~π =〈π12, . . . , π1k〉 ∈ Skn, we let πij = π1j ◦ π−11i for i, j ∈ {1, . . . , k}. Here we com-pose permutations from right to left, that is, (π1j ◦ π−11i )(x) = π1j(π−11i (x)).Note that πii = id, πij = π−1ji , and πjt ◦πij = πit hold for all i, j, t ∈ {1, . . . , k}.By an eligible ~π-tuple we mean a k-tuple ~x = 〈x1, . . . , xk〉 ∈ {0, 1, . . . , n}k suchthat πij(xi + 1) ≥ xj + 1 holds for all i, j ∈ {1, . . . , k} such that xi < n. Notethat an eligible ~π-tuple belongs to {0, 1, . . . , n−1}k∪{〈n, . . . , n〉} since xj = nimplies xi = n. The set of eligible ~π-tuples is denoted by LC(~π). It is a posetwith respect to the componentwise order: ~x ≤ ~y means that xi ≤ yi for alli ∈ {1, . . . , k}. It is trivial to check that 〈n, . . . , n〉 ∈ LC(~π) and that LC(~π) isa meet-subsemilattice of the k-th direct power of the chain {0 ≺ 1 ≺ · · · ≺ n}.Vol. 00, XX Join-distributive lattices by permutations 3Therefore, LC(~π) is a lattice, the ~π-coordinatized lattice. Its construction is mo-tivated by G. Cze´dli and E.T. Schmidt [8, Theorem 1], see also M. Stern [18],which asserts that there is a surjective cover-preserving join-homomorphismϕ : {0 ≺ · · · ≺ n}k → L, provided L is semimodular. Then, as it is easy toverify, u 7→∨{x : ϕ(x) = u} is a meet-embedding of L into {0 ≺ · · · ≺ n}k.To give an example, let π12 =(1 2 3 44 2 1 3), π13 =(1 2 3 43 2 4 1), andlet ~π = 〈π12, π13〉 ∈ S24 . Then Figure 1 also gives LC(~π); the eligible ~π-tuplesare given after the commas in the labels. For example, 23,020 in the figurecorresponds to 〈0, 2, 0〉. Note that if µ12 =(1 2 3 43 2 1 4)and µ13 = π13,then LC(~π) ∼= LC(~µ). Furthermore, the problem of characterizing those pairsof members of Skn that determine the same lattice is not solved yet if k ≥ 3.For k = 2 the solution is given in G. Cze´dli and E.T. Schmidt [10]; besidesLC(~π) ∼= LC(~µ) above, see also G. Cze´dli [7, Example 5.3] to see the difficulty.The connection of join-distributivity to LEJ(~π) and LC(~π) will be givensoon.The two constructions are equivalent. For 〈γ2, . . . , γk〉 ∈ Sk−1n , we let〈γ2, . . . , γk〉−1 = 〈γ−12 , . . . , γ−1k 〉.Proposition 1. For every ~σ ∈ Sk−1n , LEJ(~σ) is isomorphic to LC(~σ−1).In some vague sense, Figure 1 reveals why LEJ(~σ) could be of the formLC(~π) for some ~π. Namely, for x ∈ LEJ(~σ) and i ∈ {1, . . . , k}, we can definethe i-th coordinate of x as the length of the intersection of the ideal {y ∈LEJ(~σ) : y ≤ x} and the chain given in (1). However, the proof is morecomplex than this initial idea.Proof. Denote ~σ−1 by ~π = 〈π12, . . . , π1k〉. Note that π11 = σ−11 = id ∈ Sn. ForU ∈ LEJ(~σ) and i ∈ {1, . . . , k}, let U(i) = max{j : {σi(1), . . . , σi(j)} ⊆ U},where max∅ is defined to be 0. We assert that the mapϕ : LEJ(~σ)→ LC(~π), defined by U 7→ 〈U(1), . . . , U(k)〉,is a lattice isomorphism. To prove that ϕ(U) is an eligible ~π-tuple, assumethat i, j ∈ {1, . . . , k} such that U(i) < n. Then σi(U(i) + 1) /∈ U yieldsσi(U(i) + 1) /∈ {σj(1), . . . , σj(U(j))}. However, σi(U(i) + 1) ∈ {1, . . . , n} ={σj(1), . . . , σj(n)}, and we conclude that σi(U(i) + 1) = σj(t) holds for somet ∈ {U(j) + 1, . . . , n}. Henceπij(U(i) + 1) = (π1j ◦ πi1)(U(i) + 1) = π1j(πi1(U(i) + 1))= π1j(π−11i (U(i) + 1))= σ−1j(σi(U(i) + 1))= σ−1j(σj(t))= t ≥ U(j) + 1.4 K. Adaricheva and G. Cze´dli Algebra univers.This proves that ϕ(U) is an eligible ~π-tuple, and ϕ is a map from LEJ(~σ) toLC(~π). Since LEJ(~σ) is generated by the set given in (1), we concludeU =k⋃i=1{σi(1), . . . , σi(U(i))}.This implies that U is determined by 〈U(1), . . . , U(k)〉 = ϕ(U), that is, ϕ isinjective. To prove that ϕ is surjective, let ~x = 〈x1, . . . , xk〉 be a ~π-eligibletuple, that is, ~x ∈ LC(~π). DefineV =k⋃i=1{σi(1), . . . , σi(xi)}. (2)(Note that if xi = 0, then {σi(1), . . . , σi(xi)} denotes the empty set.) Forthe sake of contradiction, suppose ϕ(V ) 6= ~x. Then, by the definition of ϕ,there exists an i ∈ {1, . . . , k} such that σi(xi + 1) ∈ V . Hence, there is aj ∈ {1, . . . , k} such that σi(xi+1) ∈ {σj(1), . . . , σj(xj)}. That is, σi(xi+1) =σj(t) for some t ∈ {1, . . . , xj}. Therefore,πij(xi + 1) = π1j(πi1(xi + 1)) = π1j(π−11i (xi + 1) = σ−1j (σi(xi + 1))= σ−1j (σj(t)) = t ≤ xj ,which contradicts the ~π-eligibility of ~x. Thus ϕ(V ) = ~x and ϕ is surjective.We have shown that ϕ is bijective. For ~x ∈ LC(~π), ϕ−1(~x) is the set V givenin (2). Thus ϕ and ϕ−1 are monotone, and ϕ is a lattice isomorphism. Two descriptions. The following theorem is a straightforward consequenceof Theorems 5.1 and 5.2 in P.H. Edelman and R.E. Jamison [14], which wereformulated and proved within Combinatorics.Theorem 2. Up to isomorphism, join-distributive lattices of length n andjoin-width at most k are characterized as lattices LEJ(~σ) with ~σ ∈ Sk−1n .The next theorem was motivated and proved by the second author [7] in apurely lattice theoretical way.Theorem 3. Up to isomorphism, join-distributive lattices of length n andjoin-width at most k are characterized as the ~π-coordinatized lattices LC(~π)with ~π ∈ Sk−1n .Remark 4. Since there is no restriction on (n, k) ∈ N × {2, 3, . . .} in Theo-rems 2 and 3, one might have the feeling that, for a given n, the join-width ofa join-distributive lattice of length n can be arbitrarily large. This is not sosince, up to isomorphism, there are only finitely many join-distributive latticesof length n.The statement of Remark 4 follows from the fact that each join-distributivelattice of length n is dually isomorphic to the lattice of closed sets of a convexgeometry on the set {1, . . . , n}, see P.H. Edelman [12, Theorem 3.3] togetherVol. 00, XX Join-distributive lattices by permutations 5with the sixteenth line in the proof of Theorem 1.9 in K. Adaricheva, V.A. Gor-bunov and V.I. Tumanov [3]; see also [7, Lemma 7.4], where this is surveyed.The statement also follows, in a different way, from [7, Corollary 4.4].Remark 5. Obviously, Proposition 1 and Theorem 2 imply Theorem 3 and,similarly, Proposition 1 and Theorem 3 imply Theorem 2. Thus we obtain anew, combinatorial proof of Theorem 3 and a new, lattice theoretical proof ofTheorem 2.Comparison. We can compare Theorems 2 and 3, and the correspondingoriginal approaches, as follows.In case of Theorem 2, the construction of the lattice LEJ(~σ) is very simple,and a join-generating subset is also given.In case of Theorem 3, the elements of the lattice LC(~π) are exactly given bytheir coordinates, the eligible ~π-tuples. Moreover, the meet operation is easy,and we have a satisfactory description of the optimal meet-generating subsetsince it was proved in [7, Lemma 6.5] thatMi(LC(~π)) ={〈π11(i)− 1, . . . , π1k(i)− 1〉 : i ∈ {1, . . . , n}}.Characterization by trajectories. For a lattice L of finite length, the set{[a, b] : a ≺ b, a, b ∈ L}of prime intervals of L will be denoted by PrInt(L).For [a, b], [c, d] ∈ PrInt(L), we say that [a, b] and [c, d] are consecutive if{a, b, c, d} is a covering square, that is, a 4-element cover-preserving booleansublattice of L. The transitive reflexive closure of the consecutiveness rela-tion on PrInt(L) is an equivalence, and the blocks of this equivalence relationare called the trajectories of L; this concept was introduced for some par-ticular semimodular lattices in G. Cze´dli and E.T. Schmidt [9]. For distinct[a, b], [c, d] ∈ PrInt(L), these two prime intervals are comparable if either b ≤ c,or d ≤ a. Before formulating the last statement of the paper, it is reasonableto mention that, for any finite lattice L,L is join-distributive iff it is semimodular and meet-semidistributive. (3)This follows from K. Adaricheva, V.A. Gorbunov and V.I. Tumanov [3, The-orems 1.7 and 1.9]; see also D. Armstrong [4, Theorem 2.7] for the presentformulation.Corollary 6. For a semimodular lattice L, the following three conditions areequivalent.(i) L is join-distributive.(ii) L is of finite length, and for every trajectory T of L and every maximalchain C of L, |PrInt(C) ∩ T | = 1.(iii) L is of finite length, and no two distinct comparable prime intervals of Lbelong to the same trajectory.As an interesting consequence, note that each of (ii) and (iii) above, togetherwith semimodularity, implies that L is finite.6 K. Adaricheva and G. Cze´dli Algebra univers.Proof of Corollary 6. Since any two comparable prime intervals belong to theset of prime intervals of an appropriate maximal chain C, (ii) implies (iii). Sowe have to prove that (i) ⇒ (iii) and that (iii) ⇒ (i); we give two alternativearguments for each of these two implications. Let n = lengthL.Assume (i). Then L is semimodular by (3), and it contains no cover-preserving diamond by the definition of join-distributivity. Thus G. Cze´dli [7,Lemma 3.3] implies (ii).For a second argument, assume (i) again. Let Pow({1, . . . , n}) denote theset of all subsets of {1, . . . , n}. It is known that L is isomorphic to an appropri-ate join-subsemilattice F of the powerset(Pow({1, . . . , n});∪)such that ∅ ∈ Fand each X ∈ F \ {∅} contains an element a with the property X \ {a} ∈ F.The structure 〈{1, . . . , n};F〉 is an antimatroid on the base set {1, . . . , n} (thisconcept is due to R. E. Jamison-Waldner [16]), and the existence of an appro-priate F follows from P.H. Edelman [12, Theorem 3.3] and D. Armstrong [4,Lemma 2.5]; see also K. Adaricheva, V.A. Gorbunov and V.I. Tumanov [3,Subsection 3.1] and G. Cze´dli [7, Section 7]. Now, we can assume that L = F.We assert that, for any X,Y ∈ F,X ≺ Y iff X ⊂ Y and |Y \X | = 1. (4)The “if” part is obvious. For the sake of contradiction, suppose X ≺ Y and xand y are distinct elements in Y \X . Pick a sequence Y = Y0 ⊃ Y1 ⊃ · · · ⊃Yt = ∅ in F such that |Yi−1 \ Yi| = 1 for i ∈ {1, . . . , t}. Then there is a j suchthat |Yj ∩ {x, y}| = 1. This gives the desired contradiction since X ∪ Yj ∈ Fbut X ⊂ X ∪ Yj ⊂ Y .Armed with (4), assume that {A = B ∧ C,B,C,D = B ∪ C} is a coveringsquare in F. Note that A and B ∩ C can be different; however, A ⊆ B ∩ C.By (4), there exist u, x ∈ D such that B = D \ {u} and C = D \ {x}. Theseelements are distinct since B 6= C. Hence x ∈ B and, by A ⊆ C, x /∈ A.Using (4) again, we obtain A = B \ {x}. We have seen that whenever [A,B]and [C,D] are consecutive prime intervals, then there is a common x suchthat A = B \ {x} and C = D \ {x}. This implies that for each trajectory Tof F, there exists an xT ∈ {1, . . . , n} such that X = Y \ {xT } holds for all[X,Y ] ∈ T . Clearly, this implies that (iii) holds for F, and also for L.Next, assume (iii). Since any two prime intervals of a cover-preservingdiamond would belong to the same trajectory, L contains no such diamond.Again, there are two ways to conclude (i).First, by [7, Proposition 6.1], L is isomorphic to LC(~π) for some k and~π ∈ Sk−1n , and we obtain from Theorem 3 that (i) holds.Second, H. Abels [1, Theorem 3.9(a⇒b)] implies that L is a cover-preservingjoin-subsemilattice of a finite distributive lattice D. Thus if x ∈ L \ {1}, thenthe interval [x, x∗]L of L is a cover-preserving join-subsemilattice of D. Leta1, . . . , at be the covers of x in L, that is, the atoms of [x, x∗]L. If we had,say, a1 ≤ a2 ∨ · · · ∨ at, then we would get a contradiction in D as follows:a1 = a1 ∧ (a2 ∨ · · · ∨ at) = (a1 ∧ a2) ∨ · · · ∨ (a1 ∧ at) = x ∧ · · · ∧ x = x.Vol. 00, XX Join-distributive lattices by permutations 7Thus a1, . . . , at are independent atoms in [x, x∗]L. Therefore, it follows fromG. Gra¨tzer [15, Theorem 380] and the semimodularity of [x, x∗]L that thesublattice S generated by {a1, . . . , at} in L is the 2t-element boolean lattice. Inparticular, lengthS = t = length([x, x∗]L)since {x, x∗} ⊆ S ⊆ [x, x∗]L. Sincethe embedding is cover-preserving, the length of the interval [x, x∗]D in D isalso t. Hence |Ji ([x, x∗]D)| = t by [15, Corollary 112], which clearly implies|[x, x∗]D| ≤ 2t. Now from [x, x∗]L ⊆ [x, x∗]D and 2t = |S| ≤ |[x, x∗]L| ≤|[x, x∗]D| ≤ 2t we conclude [x, x∗]L = [x, x∗]D. This implies that [x, x∗]L isdistributive. Thus (i) holds. References[1] Abels, H.: The geometry of the chamber system of a semimodular lattice. Order 8,143–158 (1991)[2] Adaricheva, K. Representing finite convex geometries by relatively convex sets.European Journal of Combinatorics, to appear; http://arxiv.org/abs/1101.1539[3] Adaricheva, K., Gorbunov, V.A., Tumanov, V.I.: Join-semidistributive lattices andconvex geometries. Advances in Math. 173, 1–49 (2003)[4] Armstrong, D.: The Sorting Order on a Coxeter Group. Journal of CombinatorialTheory Series A 116, 1285–1305 (2009)[5] Avann, S.P.: Application of the join-irreducible excess function to semimodularlattices. Math. Annalen 142, 345–354 (1961)[6] Caspard, N., Monjardet, B.: Some lattices of closure systems on a finite set. DiscreteMathematics and Theoretical Computer Science 6, 163–190 (2004)[7] Cze´dli, G.: Coordinatization of join-distributive lattices.http://arxiv.org/abs/1208.3517[8] Cze´dli, G., Schmidt, E.T.: How to derive finite semimodular lattices from distributivelattices?. Acta Mathematica Hungarica 121, 277–282 (2008)[9] Cze´dli, G., Schmidt, E.T.: The Jordan-Ho¨lder theorem with uniqueness for groupsand semimodular lattices. Algebra Universalis 66, 69–79 (2011)[10] Cze´dli, G., Schmidt, E.T.: Intersections of composition series in groups and slimsemimodular lattices by permutations. http://arxiv.org/abs/1208.4749v1[11] Dilworth, R.P.: Lattices with unique irreducible decompositions. Annals ofMathematics (2) 41, 771–777 (1940)[12] Edelman, P.H.: Meet-distributive lattices and the anti-exchange closure. AlgebraUniversalis 10, 290–299 (1980)[13] Edelman, P.H.: Abstract convexity and meet-distributive lattices. In: Proc. Conf.Combinatorics and ordered sets (Arcata, Calif., 1985), Contemp. Math. 57, 127–150,Amer. Math. Soc., Providence, RI (1986).[14] Edelman, P.H., Jamison, R.E.: The theory of convex geometries. Geom. Dedicata 19,247–271 (1985)[15] Gra¨tzer, G.: Lattice Theory: Foundation. Birkha¨user Verlag, Basel (2011)[16] Jamison-Waldner, R.E.: Copoints in antimatroids. In: Combinatorics, graph theoryand computing, Proc. 11th southeast. Conf., Boca Raton/Florida 1980, Vol. II, Congr.Numerantium 29, 535–544 (1980)[17] Monjardet, B.: A use for frequently rediscovering a concept. Order 1, 415–417 (1985)[18] Stern, M.: Semimodular Lattices. Theory and Applications, Encyclopedia ofMathematics and its Applications 73. Cambridge University Press (1999)[19] Ward, M.: Structure Residuation. Annals of Mathematics (2) 39, 558-568 (1938)Kira Adaricheva8 K. Adaricheva and G. Cze´dli Algebra univers.Department of Mathematical Sciences, Yeshiva University, New York, 245 Lexingtonave., New York, NY 10016, USAe-mail : adariche@yu.eduGa´bor Cze´dliUniversity of Szeged, Bolyai Institute. Szeged, Aradi ve´rtanu´k tere 1, HUNGARY 6720e-mail : czedli@math.u-szeged.huURL: http://www.math.u-szeged.hu/~czedli/

Notes on the description of join-distributive lattices by permutations

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Notes on the description of join-distributive lattices by permutations

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