FINITE PSEUDOCOMPLEMENTED LATTICES ... - Semantic Scholar
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FINITE PSEUDOCOMPLEMENTED LATTICES and "PERMUTOEDRE"
C. CHAMENI NEMBUA* and B. MONJARDET** *Université de Yaoundé, **Université Paris 5 et CAMS 54 bd Raspail, F-7527O PARIS CEDEX 06, FRANCE
Abstract We study finite pseudocomplemented lattices and especially those that are also complemented. With regard to the classical results on arbitrary or distributive pseudocomplemented lattices (Glivenko, Stone, Birkhoff, Frink, Grätzer, Balbes, Horn, Varlet...), the finiteness property allows to bring significant more precise details on the structural properties of such lattices. These results can especially be applied to the lattices defined by the "weak Bruhat order" on a Coxeter group (and for instance to the lattice of permutations, called in French "le treillis permutoèdre") and to the lattice of binary bracketings. Résumé Soit T un treillis avec plus petit élément noté 0 ; l'élément t de T a un infpseudocomplément, noté g(t), si g(t) est le plus grand élément de l'ensemble des x de T tels que x∧t = 0 ; T est inf-pseudocomplémenté (IPC.) si tout élément de T a un inf-pseudocomplément. On définit dualement la notion de sup-pseudocomplément f(t) de l'élément t et de treillis suppseudocomplémenté (SPC) et par conjonction des deux propriétés IPC et SPC celle de treillis pseudocomplémenté. Ces treillis ont surtout été étudiés dans des cas où ils sont distributifs et infinis (treillis de Brouwer ou d' Heyting, treillis de Stone...). Notre intérêt pour le cas fini provient-entre autres-de ce que le "treillis permutoèdre"(Guilbaud et Rosenstiehl 1971), i.e. l'ensemble des permutations d'un ensemble fini muni de l’"ordre faible de Bruhat", est un treillis pseudo complémenté. Dans le cas d'un treillis IPC la classique correspondance de Galois associée à l'application g d' infpseudocomplémentation permet de montrer que l'ensemble des inf-pseudo compléments a une structure de treillis booléen (Frink 1962). Dans le cas fini, nous donnons d'abord une caractérisation constructive des infpseudocompléments permettant de retrouver ce résultat. Nous montrons ensuite que pour un treillis complémenté les propriétés d'être IPC ou SPC sont équivalentes. Nous décrivons ensuite de façon approfondie la structure des treillis complémentés pseudocomplémentés, ces derniers résultats s'appliquant au treillis permutoèdre et au treillis des parenthésages.
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INTRODUCTION Let L be a lattice with a least element denoted 0 ; g(t) ∈ L is a meetpseudocomplement of t ∈ L, if [x∧t = 0 if and only if x ≤ g(t)]. L is meetpseudocomplemented if every element of L has a meet-pseudocomplement. One defines dually the notion of a join-pseudocomplement f(t) of t and of a join-pseudocomplemented lattice. A lattice is pseudocomplemented if it is meet- and join-pseudocomplemented (beware! often "pseudocomplemented" means only "meet-pseudocomplemented" and a join-pseudocomplemented lattice is sometimes called a "dual pseudocomplemented" lattice). Two classes of meet-pseudocomplemented lattices have been intensively studied. First the Brouwerian (called also Heyting or implicative) lattices. They are the "relatively meet-pseudocomplemented" lattices what imply they are distributive (Glivenko 1929, Birkhoff 1940, 1948, etc....). Second, the Stone lattices which are distributive meet-pseudocomplemented lattices satisfying an additional condition (Stone 1937, Varlet 1963, Balbes and Horn 1970, etc.....). Grätzer (1978) and Varlet (1963, 1974-75) provide excellent accounts of the results known on arbitrary meet-pseudocomplemented lattices or on the above special classes. Observe that in most of the studied cases the considered lattices are infinite. We are interested here by the specific properties of the class of finite (meet- or/and join-) pseudocomplemented lattices. Indeed, the lattice of permutations (called in french, le "treillis permutoèdre", Guilbaud and Rosenstiehl 1971) and, more generally, the lattices defined by the "weak Bruhat order" on a Coxeter group (see Björner 1984) are (meet and join) pseudocomplemented lattices (Le Conte de Poly-Barbut 1990). It is also the case of the lattice of the binary bracketings (see Huang and Tamari 1972, and Lakser 1978); all these particular lattices are also complemented. In this paper we give a summary of our results on the structure of finite meet-pseudocomplemented, (meet and join) pseudocomplemented, and pseudocomplemented and complemented lattices. For a detailed account with the proofs of these results see Chameni and Monjardet 1992. The specific theory of finite meet-pseudocomplemented lattices begins with the easy but crucial observation that a (finite) lattice is meetpseudocomplemented if and only if each of its atoms has a meetpseudocomplement. So, in such lattices the meet-pseudocomplements can be expressed by means of the meet-pseudocomplements of the atoms (Theorem 1). Then one shows (Theorem 2) that the joins of atoms define a Boolean lattice isomorphic with the lattice of the meet-pseudocomplements, thus reobtaining the classical result (Frink 1962) that this last lattice is Boolean. Proposition 4 and 5 study the properties of an element in a meetpseudocomplemented lattice and especially when this element has a complement. Theorem 6 characterizes the meet-pseudocomplemented lattices which are complemented, such lattices beeing the same that the complemented join-pseudocomplemented lattices and thus that the
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complemented (and) pseudocomplemented lattices (Theorem 7). Theorem 8 gives other characterizations of such complemented and pseudocomplemented lattices and Theorem 9 summarizes all our results on the structure of such lattices. For instance, we show that the Glivenko congruence "to have the same meet-pseudocomplement" is the same that "to have the same join-pseudocomplement" or that "to have the same complements" or, etc.... (see 9.4 ). The classes of this congruence are the 2n intervals [∨A(x), ∧C(x)] with A(x) = {atoms a : a ≤ x}, C(x) = {coatoms c : c ≥ x}, and n the number of atoms -or of coatoms- of the lattice L. These results can be applied to the "concrete" lattices mentioned above. For instance, the figure below shows the lattice of permutations on 4 elements with two classes of the Glivenko congruence. 4321
4231
4312
4132
4213
1432
4123
1423
3412
1243
2143
1324
3241
2431
3214
2413
3142
1342
3421
3124
2341
2314 2134
1234
g(1423) = 3421 Glivenko class of 1423 = {1243, 1423, 4123} f(1423) = 3214 {complements of 1423} = {3214, 3241, 3421} Two (complementary) Glivenko classes of the lattice of permutations of {1,2,3,4}
RESULTS In this paper L denotes a finite lattice; 0 (respectively, 1) denotes the least (respectively, the greatest) element of L; ≤ (respectively, p , ∧ and ∨) denotes the order relation (respectively, the covering relation, the infimum or meet - operation, the supremum - or join - operation) defined on L. An element t* (respectively, t*) of L is a meet-pseudocomplement (respectively, a join-pseudocomplement) of an element t of L, if x∧t = 0 ⇔ x ≤ t* (respectively, x ∨ t = 1 ⇔ x ≥ t*). Otherwise said, t has a meet-pseudocomplement t* (respectively, a joinpseudocomplement t*) if t* (respectively, t*) is the greatest element (respectively, the least element) of the set of all elements x such that x∧t = 0 (respectively, such that x∨t = 1) .
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A lattice L is meet-pseudocomplemented (respectively, joinpseudocomplemented) if each element of L has a meet-pseudocomplement (respectively, a join-pseudocomplement). L is pseudocomplemented if it is both meet and join-pseudocomplemented (take care: often "pseudocomplemented" means only "meet-pseudocomplemented").Then, we denote by g (respectively, f) the map t → gt = t* (respectively, t → ft = t*)1. An obvious - but significant - observation made by Birkhoff (1948) is that in a meet-pseudocomplemented lattice L the map g of meetpseudocomplementation is a "symmetric Galois connection" (for the definitions of a Galois connection and of other notions of lattice theory not defined here, see Birkhoff 1967 or Grätzer 1978). Then, g3 = g, g2 = ϕ is a closure operator on L and the set G = g(L) of all meet pseudocomplements is a lattice sub meet-semilattice of L. We recall also the following classical results: g(x∨y) = gx∧gy g(x∧y) = ϕ(ϕx∧ϕy) ϕ(x∨y) = g[gx∧gy], ϕ(x∧y) = ϕx∧ϕy. An atom (respectively, a coatom of L is an element covering 0 (respectively, covered by 1). We denote by A (respectively, C) the set of all atoms (respectively, coatoms) of L, and we use the following notations: for x in L, A(x) = {a ∈ A : a ≤ x}, C(x) = {c ∈ C : c ≥ x}; A'(x) = A\A(x) ; C'(x) = C\C(x). Theorem 1 Let L be a (finite) lattice ; the two following conditions are equivalent: 1) L is a meet-pseudocomplemented lattice, 2) each atom of L has a meet-pseudocomplement (denoted by ga) These conditions imply that for all x, y in L, 3) gx = ∧g[A(x)] = g[∨A(x)], 4) ϕx = ϕ [∨A(x)] = ∧(g[A(x)]), 5) A(x) = A(ϕx), 6) gx = gy ⇔ ϕx = ϕy ⇔ A(x) = A(y). We denote by Π the equivalence defined on L by the equalities in 6) above. It is easy to see that Π is a congruence on L called the Glivenko congruence. Then, the quotient lattice L/Π is isomorphic to the lattice G of the meet-pseudocomplements of L. We denote by Πx the congruence class of x. Let A∨ = {∨X, X ⊆ A} be the set of all the join of sets of atoms of L. The set A∨ is a lattice for the order defined on L (observe that ∨∅ = 0).
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The image m(x) of an element x by a map m will be generally noted mx.
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Theorem 2 Let L be a meet-pseudocomplemented lattice: 1) The lattice A∨ of join of atoms of L is a Boolean lattice, sub joinsemilattice of L and with same least element 0. 2) g (respectively, ϕ) induces a dual isomorphism (respectively, an isomorphism) between A∨ and the lattice G of the meet-pseudocomplements of L: A(gx) = A'(x), g(x) = ϕ[∨A'(x)]. 3) The classes of the Glivenko congruence Π are the 2|A| intervals defined for each X ⊆ A by [∨X, ϕ(∨X)]: Πx = [∨A(x), ϕ[∨A(x)], Π(gx) = [∨A'(x), ϕ[∨A'(x)]. Corollary 3 The lattice G of all meet-pseudocomplements of a (finite) lattice L is a Boolean lattice, sub meet-semilattice of L, with same least and greatest elements 0 and 1. Remark. In fact, this last result is true for an arbitrary "(meet) pseudocomplemented meet-semilattice" (Frink 1962). Frink’s proof uses a concise axiomatic of a Boolean lattice, whereas Grätzer (1980) gives a direct proof of the distributivity of G. Obviously there are dual results for the join-pseudocomplemented lattices with the coatoms of L playing the role of atoms. We give now two propositions on the (meet and join) pseudocomplemented lattices preparing our results on the complemented pseudocomplemented lattices. Proposition 4 Let x be an element of the pseudocomplemented lattice L with |A| = n and |C| = p. Let us write f'x = ∨A'(x) and g'x = ∧C'(x). Then: 1) x∨y = 1 ⇒ y ≥ f'x. 2) x∧y = 0 ⇒ y ≤ g'x. 3) f'x ≤ gx∧fx, g'x ≥ gx∨fx. 4) C'(x) = g[A(x)] ⇔ gx = g'x ⇒ C(gx) = C(g'x) ⇔ C(gx) = C'(x) ⇔ C(gx) = C(fx) ⇔ |C(x)| + |C(gx)| = p. A'(x) = f[C(x)] ⇔ fx = f'x ⇒ A(fx) = A(f'x) ⇔ A(fx) = A'(x) ⇔ A(fx) = A(gx) ⇔ |A(x)| + |A(fx)| = n. Proposition 5 Let x be an element of the pseudocomplemented lattice L. The following conditions (1) or (2) imply the equivalent conditions (3) and (4): 1) gx = g'x, 2) fx = f'x.
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3) fx ≤ gx, 4) x has a complement in L. Distributive lattices are pseudocomplemented lattices since in such lattices, one easily checks that gx = ∨{t ∈ L : x∧t = 0} and fx = ∧{t ∈ L : x∨t = 1}. On the contrary, (non distributive) upper locally distributive lattices (see Monjardet 1990, for a presentation of these lattices first studied by Dilworth) are meet-pseudocomplemented lattices not pseudocomplemented. As said in the introduction, the lattices defined by the "weak Bruhat order" on a (finite) Coxeter group and the lattice of binary bracketings are pseudocomplemented and complemented lattices. So, we come now to our results on complemented pseudocomplemented lattices. First, we give six characterizations of such lattices (see also Theorem 7). Theorem 6 Let L be a meet-pseudocomplemented lattice. The following conditions are equivalent: 1) L is complemented, 2) Π1 = {1}, 3) A(x) = A ⇒ x = 1, 4) ∨A = 1, 5) g induces a bijection between A and C, 6) ϕ is the identity map on C, 7) L is strictly meet-semicomplemented (i.e., x ≠ 1 implies that there exists y ≠ 0 with x∧y = 0). Remark. In theorem 6, Condition (7) is due to Varlet (1963). In fact the significant following result shows that the complemented meet-pseudocomplemented lattices are the complemented pseudocomplemented lattices (and also the complemented joinpseudocomplemented lattices). Theorem 7 For a complemented lattice L, the three following conditions are equivalent: 1) L is meet-pseudocomplemented, 2) L is join-pseudocomplemented, 3) L is pseudocomplemented. We give now other characterizations of pseudocomplemented lattices L which are also complemented. In the following results, Π* denotes the congruence on L defined by xΠ*y ⇔ fx = fy ⇔ C(x) = C(y).
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Theorem 8 Let L be a pseudocomplemented lattice with n atoms and p coatoms. The following conditions are equivalent: 1) L is complemented, 2) For every x ∈ L, gx = g'x (or |C(x)| + |C(gx)| = p), 3) For every x ∈ L, fx = f'x (or A(x)| + |A(fx)| = n), 4) For every x ∈ L, fx ≤ gx, 5) For every x ∈ L, Π(gx) = Π*(fx), 6) Π = Π*, 7) Π0 = Π*0, 8) Π1 = Π*1. The following theorem summarizes all our results on the structure of complemented pseudocomplemented lattices ; there Ψ = f2 is the dual closure operator defined on a (join-) pseudocomplemented lattice L, F = f(L) = ΨL, and C∧ is the lattice formed by all the meet of coatoms. Theorem 9 1) f = f3 = f Ψ = Ψ f = Ψg = fϕ ≤ g = g3 = gϕ = ϕg = ϕ f = gΨ, Ψ = fg = Ψϕ ≤ ϕ = gf = ϕΨ, gx = g'x = g[∨A(x)], fx = f'x = f [∧C(x)], C'(x) = C(gx) = C(fx) = g[A(x)] , A'(x) = A(fx) = A(gx) = f[C(x)], ϕx = ∧C(x)= ϕ[∨(A(x)], Ψx = ∨A(x)= Ψ [∧C(x)]. ∧ 2) G = gL = ϕL = C is a Boolean lattice, sub meet-semilattice of L with same 0 and 1. F = fL = ΨL = A∨ is a Boolean lattice, sub join-semilattice of L, with same 0 and 1. The maps f and g (respectively, ϕ and Ψ) induce two inverse antiisomorphisms (respectively, isomorphisms) between G and F. the map g on G (respectively, f on F) is the complementation in this lattice ; gA = C = ϕC, fC = A = ΨA. 3) g (respectively, f) is a morphism of L on the dual of G (respectively, F), ϕ (respectively, Ψ) is a morphism of L on G (respectively, F). 4) For x, y ∈ L, gx = gy ⇔ ϕx = ϕy ⇔ A(x) = A(y) ⇔ ∨A(x) = ∨A(y) ⇔ fx = fy ⇔ Ψx = Ψy ⇔ C(x) = C(y) ⇔ ∧C(x) = ∧C(y) ⇔ [fx, gx] = [fy, gy] ⇔ [Ψx, ϕx] = [Ψy, ϕy]. The relation Π defined on L by these equivalent equalities is a congruence ; the congruence classes of Π are the 2n intervals [∨A(x), ∧C(x)] (with n = |A| = |C|). The maps g and f (respectively, ϕ and Ψ) are isomorphisms (respectively, involutive dual isomorphisms) between L/Π, F and G. Let (Πx)' be the class complement of the class Πx in the Boolean lattice L/Π. Then, gx = max[(Πx)'], fx = min[(Πx)'],
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ϕx = max(Πx), {complements of x in L} = (Πx)' = [fx, gx]. 5) Π1 = {1} = ∨A,
Ψx = min(Πx), Π0 = {0} = ∧C.
Note added in proof Several results of this paper were first published in the Chameni-Nembua thesis (1989, see reference below) and in a CAMS report (C. Chameni-Nembua and B. Monjardet, Les treillis pseudocomplémenté finis, Rapport CAMS P O61, Paris, 1990). Since 1991 we become aware of a M.K. Bennett and G. Birkhoff preprint (1991, to appear in Algebra Universalis) Two families of Newman lattices, of several G. Markowsky reports on the permutation (or "permutoèdre") lattice, beginning in 1990, (the last one beeing Permutation lattices revisited, August 1992, University of Maine), and of a annex (Retracts and Glivenko intervals) written with G. Markowsky in 1992 to a Birkhoff paper (to appear in the Proceedings of the 1991 Darmstadt Conference on lattice theory) ; the Bennett and G. Birkhoff paper studies a class of lattices containing both the permutation lattice and the binary bracketings lattice (especially it contains new results on this last one) ; the Markowsky reports contains old and new results on the permutation lattice (some of them have been independently got by V. Duquenne and A. Cherfouh, On the permutation lattice, Rapport CAMS P O77, Paris, 1991) ; these reports and especially the above quoted annex contain also several results which have been obtained but not published by C. Le Conte de Poly-Barbut (see reference below) and which are special cases of theorems of this paper ; indeed the C. Le Conte de Poly-Barbut results on the permutation lattice were our main motivation to study the more general class of finite pseudocomplemented (and possibly complemented) lattices. The reader will be able to find in the Chameni-Nembua and Monjardet paper Les treillis pseudocomplémentés finis (reference below) a much more complete bibliography on the permutation lattice and related topics (we just add here that the order dimension of the "multinomial lattices" and especially of the permutation lattice has been determined by S. Flath, preprint 1492, Darmstadt, 1992).
REFERENCES R. BALBES et A. HORN, Stone lattices, Duke Mathematical Journal 37 (1970), 537-546. C.LE CONTE de POLY-BARBUT, personal communications, 1986-1990. G. BIRKHOFF, Lattice Theory, American Mathematical Society, Providence, R.I. 1940, 1948, 1967 A. BJÖRNER, Orderings of Coxeter groups, in Combinatorics and algebra, C. Greene, ed., Contemporary Mathematics 34, American Mathematical Society, Providence, R.I. 175-195, 1984. C. CHAMENI-NEMBUA, Permutoèdre et choix social, thèse de doctorat de mathématiques, Université de Paris V, 1989. C. CHAMENI-NEMBUA, B. MONJARDET, Les treillis pseudocomplémenté finis, European Journal of Combinatorics 13 (1992), 89-107. O. FRINK, Pseudo-complements in semi lattices, Duke Mathematical Journal 29 (1962), 505-514 V. GLIVENKO, Sur quelques points de la logique de Brouwer, Bulletin de l’Académie des Sciences de Belgique 15 (1929), 183-188. P. GUILBAUD, P. ROSENSTIEHL, Analyse algébrique d'un scrutin, Mathématiques et Sciences humaines 4 (1963), 9-33. G. GRÄTZER, General lattice Theory, Birkhäuser Verlag, Stuttgart, 1978. S. HUANG, D. TAMARI, Problems of associativity: a simple proof for the lattice property of systems ordered by a semi-associative law, Journal of Combinatorial Theory A, 13 (1972), 7-13. H. LAKSER, Exercises 30 et 31 p. 51 in G. Grätzer, General Lattice Theory, Birkhäuser
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Verlag, Stuttgart, 1978. B. MONJARDET, The consequences of Dilworth's work for lattices with unique irreducible decompositions, in The Dilworth theorems, K. Bogart, R. Freese et J. Kung ed., Birkhäuser, Boston, 192-200, 1990. M. H. STONE, Topological representations of distributive lattices and Brouwerian logics, Casopis Pest. Mat. 67 (1937), 1-25. J. VARLET, Contributions à l'étude des treillis pseudo-complémentés et des treillis de Stone, Mémoires Société Royale des Sciences de Liège 8 (1963), 1-71. J. VARLET, Structures algébriques ordonnées, Notes d'un séminaire Université de Liège, 1974-1975 (134 pages).
Historical Note on permutoedre, Coxeter, Tamari and some other lattices (B. Monjardet, january 2012) This historical Note is a revised and (partially) updated version of the Note (in French) in Chameni Nembua and Monjardet's 1992 paper (Les treillis pseudocomplémentés finis). The references of this paper are also partially updated below, especially on finite Coxeter lattices, Tamari lattices and some generalizations of these lattices.
In 1963 Guilbaud and Rosenstiehl show that a partial order defined on the set of all linear orders on a finite set of size n (and so, equivalently on the set Sn of permutations of this set) is a lattice. Their proof –made more understandable- in the 1971 version of their paper – is taken again in Berge 1968 and in Knuth 1973 (Exercises 11 & 12, page 18-19 in the last edition). The same result also appears in the framework of rank statistics' theory. Indeed, for the need of this theory, Savage defines several orders on the set of permutations (or more generally on some sets of words). Under some probabilistic assumptions, the order defined, for example, between two permutations of the n first integers induces an order between the probabilities that these permutations be the rank vectors of n ordinal random variables. In 1964 Savage studies the possibility that these partial orders be lattices and raises several questions. Answering one of them, Yanagimoto and Okamoto (1969) define on Sn the same partial order that Guilbaud and Rosenstiehl and claim that it is a lattice (Theorem 2.1). But one must observe that they evade the only difficult point to prove by writing "it can be shown that" (!!). One can also note that their preliminary result (Proposition 2.2) amounts to showing that a partial order O contained in a linear order L has dimension 2 if and only if L is a "non separating linear extension" of O, an obvious corollary of Dushnik and Miller’ 1942 characterization of partial orders of dimension 2. The partial order on Sn considered by Guilbaud and Rosenstiehl, as well as by Yanagimoto and Okamoto, corresponds to what is often now called the "weak Bruhat order" on the symmetric group Sn (see Björner, 1984 ; note that one of the partial orders considered by Savage was the strong Bruhat order on Sn). Some simple properties of the lattice Sn are in Barbut and Monjardet (1970) and Monjardet (1971). The characterization of the irreducible elements of the lattice Sn appears in Chameni-Nembua (1989), whereas the
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characterization of the order between the join- and the meet- irreducible elements (the so-called lattice table in "formal concept analysis") is in Duquenne and Cherfouh (1991) and in Markowsky (1991). Other properties of the lattice Sn, or/and, more generally, of Bruhat weak orders on a finite Coxeter group, have been obtained by Björner (1984) who especially shows that they are orthocomplemented lattices, Le Conte de Poly-Barbut (19861990, 1994) and Leclerc (1991). Since the lattice Sn is pseudocomplemented and complemented, it also satisfies properties given in Chameni-Nembua and Monjardet (1992, 1993). A significant property is that Sn is a bounded lattice (Caspard 2000), a result generalized to any finite Coxeter lattice by Caspard, Le Conte de Poly-Barbut and Morvan (2004). On the other hand, Stanley (1984) and Edelman and Greene (1984,1987) enumerate maximal chains of the lattice Sn (or of intervals of this lattice) and show correspondences between these maximal chains and some Young tableaux. Some distributive sublattices of Sn have been first studied by Frey (1971, see also Frey and Barbut 1971). As Guilbaud observes as soon as in his 1962 paper, these sublattices are restricted domains of Sn where one can apply Condorcet's majority rule. Such subsets of Sn have been called consistent sets or acyclic domains or Condorcet domains. In this social choice context the search for Condorcet domains of maximum size has generated many works (Chameni-Nembua 1989, Abello 1981, 1985, 1987, 1991, 2004, Abello and Johnson 1984, Craven 1992, 1996, Fishburn 1992, 1996, 1997, 2002, 2005) culminating in Galambos and Reiner’s 2008 paper. See Monjardet (2009) for a survey and Danilov et al. (2012) and Danilov and Koshevoy (2012) for further results. In 1951, Tamari defines an order on the set of all possible binary bracketings on a sequence of n+1 letters. Later, he shows with Friedman (1967) and Huang (1972), that this order is a lattice Tn (see also Grätzer, 1978, Exercises 26 to 36, and Huguet 1975 for a geometrical proof of this result). These lattices Tn called Tamari lattices (and also associahedra) are shown to be pseudocomplemented and complemented by Lakser (1978). On the other hand, in 1991 Bennett and Birkhoff define a Newman (or multinomial lattice) as a lattice formed by a set of words (sequences of an alphabet) ordered from "positive elementary transformations". This class of lattices include the lattices Sn, its generalizations obtained when the words can contain more than one instance of each letter and the lattices Tn (in which the transformation corresponds to an associativity rule). Bennett and Birkhoff’s paper contains the characterization of the ordered set of the irreducible elements of the lattice Tn. Since then (or sometimes before) many classes of lattices which are restrictions or extensions of Tamari lattices -like Kreweras lattices and Stanley lattices or/and more general classes of lattices like Cambrian latticeshave been investigated. They are sets of combinatorial or geometrical objects like noncrossing partitions, bracketings, Dick words, binary trees,
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triangulations of (convex) n-gon, 321-avoiding permutations -possibly in oneto-one correspondences- and endowed with different order relations. One will find results in for instance, Adaricheva (2011), Baril and Pallo (2006), Bernardi and Bonichon (2007), Bonichon (2005), Björner and Wachs (1996, 1997), Chapoton (2006), Dehornoy (2010, 2011), Edelman (1980), Edelman and Reiner (1996), Edelman and Simion (1994), Flath (1993, 1994), Geyer (1994), Kreweras (1972), Lee (1989), Pallo (1986, 1987, 1990, 1993, 2000, 2003), Reading (2004, 2006, 2012), Santocanale (2007), Santocanale and Wehrung (2011) Simion and Ullman (1991) or Sünic (2007). REFERENCES K. ADARICHEVA, Stasheff polytope as a sublattice of a permutohedron [http://arxiv.org/abs/1101.1536] K. ADARICHEVA, Almost Distributive Lattices in Connection to Permutation Lattices, unpublished notes, June 11 2011. D. ARMSTRONG Generalized Noncrossing Partitions and Combinatorics of Coxeter Groups. Memoirs of the American Mathematical Society, Volume 202 • Number 949, 2009. R. BALBES, A. HORN (1970) Stone lattices, Duke Mathematical Journal 37, 537-546. R. BALBES, A. HORN (1970) Injective and projective Heyting Algebras, Transactions of the American Mathematical Society 148, 549-559. M. BARBUT, B. MONJARDET, Ordre et Classification, Algèbre et Combinatoire, Hachette, Paris, 1970. J.L. BARIL, J.M. PALLO (2006) Efficient lower and upper bounds of the diagonal-flip distance between triangulations, Information Processing Letters 100, 131-136. J.L. BARIL, J.M. PALLO (2006) The pahagocyte lattice of Dick words, Order 23, 97-107. M.K. BENNETT, G. BIRKHOFF, Two families of Newman lattices, report, April 1991. M.K. BENNETT, G. BIRKHOFF (1994) Two families of Newman lattices, Algebra Universalis 32 (1), 115-144. Cl. BERGE, Eléments de combinatoire, Dunod, Paris, 1968, pp.114-117 (Principles of Combinatorics, Academic Press, 1971). O. BERNARDI, N. BONICHON (2007) Catalan's intervals and realizers of triangulations, in Proceedings of the 19th International Conference on Formal Power Series and Algebraic Combinatorics, Nankai University, Tianjin, China (15 pages, http://igm.univmlv.fr/~fpsac/FPSAC07/contrib_papers.html). G. BIRKHOFF, Lattice Theory, American Mathematical Society, Providence, R.I. 1940, 1948, 1967. A. BJÖRNER (1984) Orderings of Coxeter groups, in Combinatorics and Algebra, ed. C. Greene, Contemporary Mathematics 34, American Mathematical Society, Providence, R.I. pp.175-195. A. BJÖRNER, M. L. WACHS (1996) Shellable Nonpure Complexes and Posets, Transactions of the American Mathematical Society 348, 1299-1327. A. BJÖRNER, M. L. WACHS (1997) Shellable non-pure complexes and posets. II Transactions of the American Mathematical Society 397, 3945–3975. N. BONICHON (2005) A bijection between realizers of maximal plane graphs and pairs of non-crossing Dyck paths, Discrete Mathematics 298, 104-114. N. CASPARD (1999) A characterization for all interval doubling schemes of the lattice of
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P.C. FISHBURN (1996) Decision theory and discrete mathematics, Discrete Applied Mathematics 68(3), 209-221. P.C. FISHBURN (1997) Acyclic sets of linear orders, Social Choice and Welfare 14, 113124. P.C. FISHBURN (2002) Acyclic sets of linear orders: A progress report, Social Choice and Welfare 19(2), 113-124. P.C. FISHBURN (2005) Subcyclic sets of linear orders, Social Choice and Welfare 24(2), 199-210. S. FLATH (1993) The order dimension of multinomial lattices, Order 10, 201-219 S. FLATH, Multinomial lattices, An approach to the structure of multipermutations via formal concept analysis, Doctoral Dissertation Technischen Hochschule Darmstadt, 1994. L. FREY, Parties distributives du treillis des permutations, in Ordres totaux finis, Mathématiques et Sciences de l'Homme XII, Paris, Gauthier-Villars, pp. 115-126, 1971. L. FREY, M. BARBUT, Techniques ordinales en analyse des données. Algèbre et Combinatoire, Hachette, Paris, 1971. H. FRIEDMAN, D. TAMARI (1967) Une structure de treillis fini induite par une loi demi-associative, Journal of Combinatorial Theory A 2, 215-242. O. FRINK (1962) Pseudo-complements in semi-lattices, Duke Mathematical Journal 29, 505-514. V. GEYER (1994) On Tamari lattices, Discrete Mathematics 133, 99-122. V. GLIVENKO (1929) Sur quelques points de la logique de Brouwer, Bulletin Académie des Sciences de Belgique 15, 183-188. G. Th. GUILBAUD (1952) Les théories de l'intérêt général et le problème logique de l'agrégation, Economie Appliquée 5(4), 1952, reprinted in Éléments de la théorie des jeux, Dunod, Paris, 1968. G. Th. GUILBAUD et P. ROSENSTIEHL (1963) Analyse algébrique d'un scrutin, Mathématiques et Sciences humaines 4, 9-33, and in Ordres totaux finis, Mathématiques et Sciences de l'Homme XII, Paris, Gauthier-Villars, pp. 71-100, 1971. G. GRÄTZER General lattice Theory, Birkhäuser Verlag, Stuttgart, 1978. S. HUANG, D. TAMARI (1972) Problems of associativity: a simple proof for the lattice property of systems ordered by a semi-associative law, Journal of Combinatorial Theory A 13, 7-13. D. HUGUET (1975) La structure du treillis des polyèdres de parenthésage, Algebra Universalis 5, 82-87. C. R. JOHNSON Remarks on mathematical social choice, Working paper 78-25, Dept of Economics, University of Maryland, College Park, 1978. T. KATRINAK (1973) The structure of distributive double p-algebras, regularity and congruences, Algebra Universalis 3, 238-246. D.E. KNUTH "Sorting and searching", The Art of Computer Programming, vol.3, Reading Mass., Addison Wesley, 1973, 2d edition 1998. H. LAKSER (1978) Exercises 30 et 31 p. 51 in G. Grätzer, General lattice Theory, first edition, Birkhäuser Verlag, Stuttgart (p.68 in second edition, 1998). B. LECLERC (1994) A finite Coxeter group whose weak Bruhat order is not symmetric chain, European Journal of Combinatorics 15, 181-185. C. LE CONTE de POLY-BARBUT Personal communications, 1986-1990. C. LE CONTE de POLY-BARBUT (1990) Automorphismes du permutoèdre et votes de Condorcet, Mathématiques, Informatique et Sciences humaines 111, 73-82. C. LE CONTE de POLY-BARBUT (1990) Le diagramme du treillis permutoèdre est intersection des diagrammes de deux produits directs d'ordres totaux, Mathématiques,
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FINITE PSEUDOCOMPLEMENTED LATTICES and "PERMUTOEDRE"
C. CHAMENI NEMBUA* and B. MONJARDET** *Université de Yaoundé, **Université Paris 5 et CAMS 54 bd Raspail, F-7527O PARIS CEDEX 06, FRANCE
Abstract We study finite pseudocomplemented lattices and especially those that are also complemented. With regard to the classical results on arbitrary or distributive pseudocomplemented lattices (Glivenko, Stone, Birkhoff, Frink, Grätzer, Balbes, Horn, Varlet...), the finiteness property allows to bring significant more precise details on the structural properties of such lattices. These results can especially be applied to the lattices defined by the "weak Bruhat order" on a Coxeter group (and for instance to the lattice of permutations, called in French "le treillis permutoèdre") and to the lattice of binary bracketings. Résumé Soit T un treillis avec plus petit élément noté 0 ; l'élément t de T a un infpseudocomplément, noté g(t), si g(t) est le plus grand élément de l'ensemble des x de T tels que x∧t = 0 ; T est inf-pseudocomplémenté (IPC.) si tout élément de T a un inf-pseudocomplément. On définit dualement la notion de sup-pseudocomplément f(t) de l'élément t et de treillis suppseudocomplémenté (SPC) et par conjonction des deux propriétés IPC et SPC celle de treillis pseudocomplémenté. Ces treillis ont surtout été étudiés dans des cas où ils sont distributifs et infinis (treillis de Brouwer ou d' Heyting, treillis de Stone...). Notre intérêt pour le cas fini provient-entre autres-de ce que le "treillis permutoèdre"(Guilbaud et Rosenstiehl 1971), i.e. l'ensemble des permutations d'un ensemble fini muni de l’"ordre faible de Bruhat", est un treillis pseudo complémenté. Dans le cas d'un treillis IPC la classique correspondance de Galois associée à l'application g d' infpseudocomplémentation permet de montrer que l'ensemble des inf-pseudo compléments a une structure de treillis booléen (Frink 1962). Dans le cas fini, nous donnons d'abord une caractérisation constructive des infpseudocompléments permettant de retrouver ce résultat. Nous montrons ensuite que pour un treillis complémenté les propriétés d'être IPC ou SPC sont équivalentes. Nous décrivons ensuite de façon approfondie la structure des treillis complémentés pseudocomplémentés, ces derniers résultats s'appliquant au treillis permutoèdre et au treillis des parenthésages.
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INTRODUCTION Let L be a lattice with a least element denoted 0 ; g(t) ∈ L is a meetpseudocomplement of t ∈ L, if [x∧t = 0 if and only if x ≤ g(t)]. L is meetpseudocomplemented if every element of L has a meet-pseudocomplement. One defines dually the notion of a join-pseudocomplement f(t) of t and of a join-pseudocomplemented lattice. A lattice is pseudocomplemented if it is meet- and join-pseudocomplemented (beware! often "pseudocomplemented" means only "meet-pseudocomplemented" and a join-pseudocomplemented lattice is sometimes called a "dual pseudocomplemented" lattice). Two classes of meet-pseudocomplemented lattices have been intensively studied. First the Brouwerian (called also Heyting or implicative) lattices. They are the "relatively meet-pseudocomplemented" lattices what imply they are distributive (Glivenko 1929, Birkhoff 1940, 1948, etc....). Second, the Stone lattices which are distributive meet-pseudocomplemented lattices satisfying an additional condition (Stone 1937, Varlet 1963, Balbes and Horn 1970, etc.....). Grätzer (1978) and Varlet (1963, 1974-75) provide excellent accounts of the results known on arbitrary meet-pseudocomplemented lattices or on the above special classes. Observe that in most of the studied cases the considered lattices are infinite. We are interested here by the specific properties of the class of finite (meet- or/and join-) pseudocomplemented lattices. Indeed, the lattice of permutations (called in french, le "treillis permutoèdre", Guilbaud and Rosenstiehl 1971) and, more generally, the lattices defined by the "weak Bruhat order" on a Coxeter group (see Björner 1984) are (meet and join) pseudocomplemented lattices (Le Conte de Poly-Barbut 1990). It is also the case of the lattice of the binary bracketings (see Huang and Tamari 1972, and Lakser 1978); all these particular lattices are also complemented. In this paper we give a summary of our results on the structure of finite meet-pseudocomplemented, (meet and join) pseudocomplemented, and pseudocomplemented and complemented lattices. For a detailed account with the proofs of these results see Chameni and Monjardet 1992. The specific theory of finite meet-pseudocomplemented lattices begins with the easy but crucial observation that a (finite) lattice is meetpseudocomplemented if and only if each of its atoms has a meetpseudocomplement. So, in such lattices the meet-pseudocomplements can be expressed by means of the meet-pseudocomplements of the atoms (Theorem 1). Then one shows (Theorem 2) that the joins of atoms define a Boolean lattice isomorphic with the lattice of the meet-pseudocomplements, thus reobtaining the classical result (Frink 1962) that this last lattice is Boolean. Proposition 4 and 5 study the properties of an element in a meetpseudocomplemented lattice and especially when this element has a complement. Theorem 6 characterizes the meet-pseudocomplemented lattices which are complemented, such lattices beeing the same that the complemented join-pseudocomplemented lattices and thus that the
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complemented (and) pseudocomplemented lattices (Theorem 7). Theorem 8 gives other characterizations of such complemented and pseudocomplemented lattices and Theorem 9 summarizes all our results on the structure of such lattices. For instance, we show that the Glivenko congruence "to have the same meet-pseudocomplement" is the same that "to have the same join-pseudocomplement" or that "to have the same complements" or, etc.... (see 9.4 ). The classes of this congruence are the 2n intervals [∨A(x), ∧C(x)] with A(x) = {atoms a : a ≤ x}, C(x) = {coatoms c : c ≥ x}, and n the number of atoms -or of coatoms- of the lattice L. These results can be applied to the "concrete" lattices mentioned above. For instance, the figure below shows the lattice of permutations on 4 elements with two classes of the Glivenko congruence. 4321
4231
4312
4132
4213
1432
4123
1423
3412
1243
2143
1324
3241
2431
3214
2413
3142
1342
3421
3124
2341
2314 2134
1234
g(1423) = 3421 Glivenko class of 1423 = {1243, 1423, 4123} f(1423) = 3214 {complements of 1423} = {3214, 3241, 3421} Two (complementary) Glivenko classes of the lattice of permutations of {1,2,3,4}
RESULTS In this paper L denotes a finite lattice; 0 (respectively, 1) denotes the least (respectively, the greatest) element of L; ≤ (respectively, p , ∧ and ∨) denotes the order relation (respectively, the covering relation, the infimum or meet - operation, the supremum - or join - operation) defined on L. An element t* (respectively, t*) of L is a meet-pseudocomplement (respectively, a join-pseudocomplement) of an element t of L, if x∧t = 0 ⇔ x ≤ t* (respectively, x ∨ t = 1 ⇔ x ≥ t*). Otherwise said, t has a meet-pseudocomplement t* (respectively, a joinpseudocomplement t*) if t* (respectively, t*) is the greatest element (respectively, the least element) of the set of all elements x such that x∧t = 0 (respectively, such that x∨t = 1) .
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A lattice L is meet-pseudocomplemented (respectively, joinpseudocomplemented) if each element of L has a meet-pseudocomplement (respectively, a join-pseudocomplement). L is pseudocomplemented if it is both meet and join-pseudocomplemented (take care: often "pseudocomplemented" means only "meet-pseudocomplemented").Then, we denote by g (respectively, f) the map t → gt = t* (respectively, t → ft = t*)1. An obvious - but significant - observation made by Birkhoff (1948) is that in a meet-pseudocomplemented lattice L the map g of meetpseudocomplementation is a "symmetric Galois connection" (for the definitions of a Galois connection and of other notions of lattice theory not defined here, see Birkhoff 1967 or Grätzer 1978). Then, g3 = g, g2 = ϕ is a closure operator on L and the set G = g(L) of all meet pseudocomplements is a lattice sub meet-semilattice of L. We recall also the following classical results: g(x∨y) = gx∧gy g(x∧y) = ϕ(ϕx∧ϕy) ϕ(x∨y) = g[gx∧gy], ϕ(x∧y) = ϕx∧ϕy. An atom (respectively, a coatom of L is an element covering 0 (respectively, covered by 1). We denote by A (respectively, C) the set of all atoms (respectively, coatoms) of L, and we use the following notations: for x in L, A(x) = {a ∈ A : a ≤ x}, C(x) = {c ∈ C : c ≥ x}; A'(x) = A\A(x) ; C'(x) = C\C(x). Theorem 1 Let L be a (finite) lattice ; the two following conditions are equivalent: 1) L is a meet-pseudocomplemented lattice, 2) each atom of L has a meet-pseudocomplement (denoted by ga) These conditions imply that for all x, y in L, 3) gx = ∧g[A(x)] = g[∨A(x)], 4) ϕx = ϕ [∨A(x)] = ∧(g[A(x)]), 5) A(x) = A(ϕx), 6) gx = gy ⇔ ϕx = ϕy ⇔ A(x) = A(y). We denote by Π the equivalence defined on L by the equalities in 6) above. It is easy to see that Π is a congruence on L called the Glivenko congruence. Then, the quotient lattice L/Π is isomorphic to the lattice G of the meet-pseudocomplements of L. We denote by Πx the congruence class of x. Let A∨ = {∨X, X ⊆ A} be the set of all the join of sets of atoms of L. The set A∨ is a lattice for the order defined on L (observe that ∨∅ = 0).
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The image m(x) of an element x by a map m will be generally noted mx.
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Theorem 2 Let L be a meet-pseudocomplemented lattice: 1) The lattice A∨ of join of atoms of L is a Boolean lattice, sub joinsemilattice of L and with same least element 0. 2) g (respectively, ϕ) induces a dual isomorphism (respectively, an isomorphism) between A∨ and the lattice G of the meet-pseudocomplements of L: A(gx) = A'(x), g(x) = ϕ[∨A'(x)]. 3) The classes of the Glivenko congruence Π are the 2|A| intervals defined for each X ⊆ A by [∨X, ϕ(∨X)]: Πx = [∨A(x), ϕ[∨A(x)], Π(gx) = [∨A'(x), ϕ[∨A'(x)]. Corollary 3 The lattice G of all meet-pseudocomplements of a (finite) lattice L is a Boolean lattice, sub meet-semilattice of L, with same least and greatest elements 0 and 1. Remark. In fact, this last result is true for an arbitrary "(meet) pseudocomplemented meet-semilattice" (Frink 1962). Frink’s proof uses a concise axiomatic of a Boolean lattice, whereas Grätzer (1980) gives a direct proof of the distributivity of G. Obviously there are dual results for the join-pseudocomplemented lattices with the coatoms of L playing the role of atoms. We give now two propositions on the (meet and join) pseudocomplemented lattices preparing our results on the complemented pseudocomplemented lattices. Proposition 4 Let x be an element of the pseudocomplemented lattice L with |A| = n and |C| = p. Let us write f'x = ∨A'(x) and g'x = ∧C'(x). Then: 1) x∨y = 1 ⇒ y ≥ f'x. 2) x∧y = 0 ⇒ y ≤ g'x. 3) f'x ≤ gx∧fx, g'x ≥ gx∨fx. 4) C'(x) = g[A(x)] ⇔ gx = g'x ⇒ C(gx) = C(g'x) ⇔ C(gx) = C'(x) ⇔ C(gx) = C(fx) ⇔ |C(x)| + |C(gx)| = p. A'(x) = f[C(x)] ⇔ fx = f'x ⇒ A(fx) = A(f'x) ⇔ A(fx) = A'(x) ⇔ A(fx) = A(gx) ⇔ |A(x)| + |A(fx)| = n. Proposition 5 Let x be an element of the pseudocomplemented lattice L. The following conditions (1) or (2) imply the equivalent conditions (3) and (4): 1) gx = g'x, 2) fx = f'x.
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3) fx ≤ gx, 4) x has a complement in L. Distributive lattices are pseudocomplemented lattices since in such lattices, one easily checks that gx = ∨{t ∈ L : x∧t = 0} and fx = ∧{t ∈ L : x∨t = 1}. On the contrary, (non distributive) upper locally distributive lattices (see Monjardet 1990, for a presentation of these lattices first studied by Dilworth) are meet-pseudocomplemented lattices not pseudocomplemented. As said in the introduction, the lattices defined by the "weak Bruhat order" on a (finite) Coxeter group and the lattice of binary bracketings are pseudocomplemented and complemented lattices. So, we come now to our results on complemented pseudocomplemented lattices. First, we give six characterizations of such lattices (see also Theorem 7). Theorem 6 Let L be a meet-pseudocomplemented lattice. The following conditions are equivalent: 1) L is complemented, 2) Π1 = {1}, 3) A(x) = A ⇒ x = 1, 4) ∨A = 1, 5) g induces a bijection between A and C, 6) ϕ is the identity map on C, 7) L is strictly meet-semicomplemented (i.e., x ≠ 1 implies that there exists y ≠ 0 with x∧y = 0). Remark. In theorem 6, Condition (7) is due to Varlet (1963). In fact the significant following result shows that the complemented meet-pseudocomplemented lattices are the complemented pseudocomplemented lattices (and also the complemented joinpseudocomplemented lattices). Theorem 7 For a complemented lattice L, the three following conditions are equivalent: 1) L is meet-pseudocomplemented, 2) L is join-pseudocomplemented, 3) L is pseudocomplemented. We give now other characterizations of pseudocomplemented lattices L which are also complemented. In the following results, Π* denotes the congruence on L defined by xΠ*y ⇔ fx = fy ⇔ C(x) = C(y).
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Theorem 8 Let L be a pseudocomplemented lattice with n atoms and p coatoms. The following conditions are equivalent: 1) L is complemented, 2) For every x ∈ L, gx = g'x (or |C(x)| + |C(gx)| = p), 3) For every x ∈ L, fx = f'x (or A(x)| + |A(fx)| = n), 4) For every x ∈ L, fx ≤ gx, 5) For every x ∈ L, Π(gx) = Π*(fx), 6) Π = Π*, 7) Π0 = Π*0, 8) Π1 = Π*1. The following theorem summarizes all our results on the structure of complemented pseudocomplemented lattices ; there Ψ = f2 is the dual closure operator defined on a (join-) pseudocomplemented lattice L, F = f(L) = ΨL, and C∧ is the lattice formed by all the meet of coatoms. Theorem 9 1) f = f3 = f Ψ = Ψ f = Ψg = fϕ ≤ g = g3 = gϕ = ϕg = ϕ f = gΨ, Ψ = fg = Ψϕ ≤ ϕ = gf = ϕΨ, gx = g'x = g[∨A(x)], fx = f'x = f [∧C(x)], C'(x) = C(gx) = C(fx) = g[A(x)] , A'(x) = A(fx) = A(gx) = f[C(x)], ϕx = ∧C(x)= ϕ[∨(A(x)], Ψx = ∨A(x)= Ψ [∧C(x)]. ∧ 2) G = gL = ϕL = C is a Boolean lattice, sub meet-semilattice of L with same 0 and 1. F = fL = ΨL = A∨ is a Boolean lattice, sub join-semilattice of L, with same 0 and 1. The maps f and g (respectively, ϕ and Ψ) induce two inverse antiisomorphisms (respectively, isomorphisms) between G and F. the map g on G (respectively, f on F) is the complementation in this lattice ; gA = C = ϕC, fC = A = ΨA. 3) g (respectively, f) is a morphism of L on the dual of G (respectively, F), ϕ (respectively, Ψ) is a morphism of L on G (respectively, F). 4) For x, y ∈ L, gx = gy ⇔ ϕx = ϕy ⇔ A(x) = A(y) ⇔ ∨A(x) = ∨A(y) ⇔ fx = fy ⇔ Ψx = Ψy ⇔ C(x) = C(y) ⇔ ∧C(x) = ∧C(y) ⇔ [fx, gx] = [fy, gy] ⇔ [Ψx, ϕx] = [Ψy, ϕy]. The relation Π defined on L by these equivalent equalities is a congruence ; the congruence classes of Π are the 2n intervals [∨A(x), ∧C(x)] (with n = |A| = |C|). The maps g and f (respectively, ϕ and Ψ) are isomorphisms (respectively, involutive dual isomorphisms) between L/Π, F and G. Let (Πx)' be the class complement of the class Πx in the Boolean lattice L/Π. Then, gx = max[(Πx)'], fx = min[(Πx)'],
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ϕx = max(Πx), {complements of x in L} = (Πx)' = [fx, gx]. 5) Π1 = {1} = ∨A,
Ψx = min(Πx), Π0 = {0} = ∧C.
Note added in proof Several results of this paper were first published in the Chameni-Nembua thesis (1989, see reference below) and in a CAMS report (C. Chameni-Nembua and B. Monjardet, Les treillis pseudocomplémenté finis, Rapport CAMS P O61, Paris, 1990). Since 1991 we become aware of a M.K. Bennett and G. Birkhoff preprint (1991, to appear in Algebra Universalis) Two families of Newman lattices, of several G. Markowsky reports on the permutation (or "permutoèdre") lattice, beginning in 1990, (the last one beeing Permutation lattices revisited, August 1992, University of Maine), and of a annex (Retracts and Glivenko intervals) written with G. Markowsky in 1992 to a Birkhoff paper (to appear in the Proceedings of the 1991 Darmstadt Conference on lattice theory) ; the Bennett and G. Birkhoff paper studies a class of lattices containing both the permutation lattice and the binary bracketings lattice (especially it contains new results on this last one) ; the Markowsky reports contains old and new results on the permutation lattice (some of them have been independently got by V. Duquenne and A. Cherfouh, On the permutation lattice, Rapport CAMS P O77, Paris, 1991) ; these reports and especially the above quoted annex contain also several results which have been obtained but not published by C. Le Conte de Poly-Barbut (see reference below) and which are special cases of theorems of this paper ; indeed the C. Le Conte de Poly-Barbut results on the permutation lattice were our main motivation to study the more general class of finite pseudocomplemented (and possibly complemented) lattices. The reader will be able to find in the Chameni-Nembua and Monjardet paper Les treillis pseudocomplémentés finis (reference below) a much more complete bibliography on the permutation lattice and related topics (we just add here that the order dimension of the "multinomial lattices" and especially of the permutation lattice has been determined by S. Flath, preprint 1492, Darmstadt, 1992).
REFERENCES R. BALBES et A. HORN, Stone lattices, Duke Mathematical Journal 37 (1970), 537-546. C.LE CONTE de POLY-BARBUT, personal communications, 1986-1990. G. BIRKHOFF, Lattice Theory, American Mathematical Society, Providence, R.I. 1940, 1948, 1967 A. BJÖRNER, Orderings of Coxeter groups, in Combinatorics and algebra, C. Greene, ed., Contemporary Mathematics 34, American Mathematical Society, Providence, R.I. 175-195, 1984. C. CHAMENI-NEMBUA, Permutoèdre et choix social, thèse de doctorat de mathématiques, Université de Paris V, 1989. C. CHAMENI-NEMBUA, B. MONJARDET, Les treillis pseudocomplémenté finis, European Journal of Combinatorics 13 (1992), 89-107. O. FRINK, Pseudo-complements in semi lattices, Duke Mathematical Journal 29 (1962), 505-514 V. GLIVENKO, Sur quelques points de la logique de Brouwer, Bulletin de l’Académie des Sciences de Belgique 15 (1929), 183-188. P. GUILBAUD, P. ROSENSTIEHL, Analyse algébrique d'un scrutin, Mathématiques et Sciences humaines 4 (1963), 9-33. G. GRÄTZER, General lattice Theory, Birkhäuser Verlag, Stuttgart, 1978. S. HUANG, D. TAMARI, Problems of associativity: a simple proof for the lattice property of systems ordered by a semi-associative law, Journal of Combinatorial Theory A, 13 (1972), 7-13. H. LAKSER, Exercises 30 et 31 p. 51 in G. Grätzer, General Lattice Theory, Birkhäuser
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Verlag, Stuttgart, 1978. B. MONJARDET, The consequences of Dilworth's work for lattices with unique irreducible decompositions, in The Dilworth theorems, K. Bogart, R. Freese et J. Kung ed., Birkhäuser, Boston, 192-200, 1990. M. H. STONE, Topological representations of distributive lattices and Brouwerian logics, Casopis Pest. Mat. 67 (1937), 1-25. J. VARLET, Contributions à l'étude des treillis pseudo-complémentés et des treillis de Stone, Mémoires Société Royale des Sciences de Liège 8 (1963), 1-71. J. VARLET, Structures algébriques ordonnées, Notes d'un séminaire Université de Liège, 1974-1975 (134 pages).
Historical Note on permutoedre, Coxeter, Tamari and some other lattices (B. Monjardet, january 2012) This historical Note is a revised and (partially) updated version of the Note (in French) in Chameni Nembua and Monjardet's 1992 paper (Les treillis pseudocomplémentés finis). The references of this paper are also partially updated below, especially on finite Coxeter lattices, Tamari lattices and some generalizations of these lattices.
In 1963 Guilbaud and Rosenstiehl show that a partial order defined on the set of all linear orders on a finite set of size n (and so, equivalently on the set Sn of permutations of this set) is a lattice. Their proof –made more understandable- in the 1971 version of their paper – is taken again in Berge 1968 and in Knuth 1973 (Exercises 11 & 12, page 18-19 in the last edition). The same result also appears in the framework of rank statistics' theory. Indeed, for the need of this theory, Savage defines several orders on the set of permutations (or more generally on some sets of words). Under some probabilistic assumptions, the order defined, for example, between two permutations of the n first integers induces an order between the probabilities that these permutations be the rank vectors of n ordinal random variables. In 1964 Savage studies the possibility that these partial orders be lattices and raises several questions. Answering one of them, Yanagimoto and Okamoto (1969) define on Sn the same partial order that Guilbaud and Rosenstiehl and claim that it is a lattice (Theorem 2.1). But one must observe that they evade the only difficult point to prove by writing "it can be shown that" (!!). One can also note that their preliminary result (Proposition 2.2) amounts to showing that a partial order O contained in a linear order L has dimension 2 if and only if L is a "non separating linear extension" of O, an obvious corollary of Dushnik and Miller’ 1942 characterization of partial orders of dimension 2. The partial order on Sn considered by Guilbaud and Rosenstiehl, as well as by Yanagimoto and Okamoto, corresponds to what is often now called the "weak Bruhat order" on the symmetric group Sn (see Björner, 1984 ; note that one of the partial orders considered by Savage was the strong Bruhat order on Sn). Some simple properties of the lattice Sn are in Barbut and Monjardet (1970) and Monjardet (1971). The characterization of the irreducible elements of the lattice Sn appears in Chameni-Nembua (1989), whereas the
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characterization of the order between the join- and the meet- irreducible elements (the so-called lattice table in "formal concept analysis") is in Duquenne and Cherfouh (1991) and in Markowsky (1991). Other properties of the lattice Sn, or/and, more generally, of Bruhat weak orders on a finite Coxeter group, have been obtained by Björner (1984) who especially shows that they are orthocomplemented lattices, Le Conte de Poly-Barbut (19861990, 1994) and Leclerc (1991). Since the lattice Sn is pseudocomplemented and complemented, it also satisfies properties given in Chameni-Nembua and Monjardet (1992, 1993). A significant property is that Sn is a bounded lattice (Caspard 2000), a result generalized to any finite Coxeter lattice by Caspard, Le Conte de Poly-Barbut and Morvan (2004). On the other hand, Stanley (1984) and Edelman and Greene (1984,1987) enumerate maximal chains of the lattice Sn (or of intervals of this lattice) and show correspondences between these maximal chains and some Young tableaux. Some distributive sublattices of Sn have been first studied by Frey (1971, see also Frey and Barbut 1971). As Guilbaud observes as soon as in his 1962 paper, these sublattices are restricted domains of Sn where one can apply Condorcet's majority rule. Such subsets of Sn have been called consistent sets or acyclic domains or Condorcet domains. In this social choice context the search for Condorcet domains of maximum size has generated many works (Chameni-Nembua 1989, Abello 1981, 1985, 1987, 1991, 2004, Abello and Johnson 1984, Craven 1992, 1996, Fishburn 1992, 1996, 1997, 2002, 2005) culminating in Galambos and Reiner’s 2008 paper. See Monjardet (2009) for a survey and Danilov et al. (2012) and Danilov and Koshevoy (2012) for further results. In 1951, Tamari defines an order on the set of all possible binary bracketings on a sequence of n+1 letters. Later, he shows with Friedman (1967) and Huang (1972), that this order is a lattice Tn (see also Grätzer, 1978, Exercises 26 to 36, and Huguet 1975 for a geometrical proof of this result). These lattices Tn called Tamari lattices (and also associahedra) are shown to be pseudocomplemented and complemented by Lakser (1978). On the other hand, in 1991 Bennett and Birkhoff define a Newman (or multinomial lattice) as a lattice formed by a set of words (sequences of an alphabet) ordered from "positive elementary transformations". This class of lattices include the lattices Sn, its generalizations obtained when the words can contain more than one instance of each letter and the lattices Tn (in which the transformation corresponds to an associativity rule). Bennett and Birkhoff’s paper contains the characterization of the ordered set of the irreducible elements of the lattice Tn. Since then (or sometimes before) many classes of lattices which are restrictions or extensions of Tamari lattices -like Kreweras lattices and Stanley lattices or/and more general classes of lattices like Cambrian latticeshave been investigated. They are sets of combinatorial or geometrical objects like noncrossing partitions, bracketings, Dick words, binary trees,
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triangulations of (convex) n-gon, 321-avoiding permutations -possibly in oneto-one correspondences- and endowed with different order relations. One will find results in for instance, Adaricheva (2011), Baril and Pallo (2006), Bernardi and Bonichon (2007), Bonichon (2005), Björner and Wachs (1996, 1997), Chapoton (2006), Dehornoy (2010, 2011), Edelman (1980), Edelman and Reiner (1996), Edelman and Simion (1994), Flath (1993, 1994), Geyer (1994), Kreweras (1972), Lee (1989), Pallo (1986, 1987, 1990, 1993, 2000, 2003), Reading (2004, 2006, 2012), Santocanale (2007), Santocanale and Wehrung (2011) Simion and Ullman (1991) or Sünic (2007). REFERENCES K. ADARICHEVA, Stasheff polytope as a sublattice of a permutohedron [http://arxiv.org/abs/1101.1536] K. ADARICHEVA, Almost Distributive Lattices in Connection to Permutation Lattices, unpublished notes, June 11 2011. D. ARMSTRONG Generalized Noncrossing Partitions and Combinatorics of Coxeter Groups. Memoirs of the American Mathematical Society, Volume 202 • Number 949, 2009. R. BALBES, A. HORN (1970) Stone lattices, Duke Mathematical Journal 37, 537-546. R. BALBES, A. HORN (1970) Injective and projective Heyting Algebras, Transactions of the American Mathematical Society 148, 549-559. M. BARBUT, B. MONJARDET, Ordre et Classification, Algèbre et Combinatoire, Hachette, Paris, 1970. J.L. BARIL, J.M. PALLO (2006) Efficient lower and upper bounds of the diagonal-flip distance between triangulations, Information Processing Letters 100, 131-136. J.L. BARIL, J.M. PALLO (2006) The pahagocyte lattice of Dick words, Order 23, 97-107. M.K. BENNETT, G. BIRKHOFF, Two families of Newman lattices, report, April 1991. M.K. BENNETT, G. BIRKHOFF (1994) Two families of Newman lattices, Algebra Universalis 32 (1), 115-144. Cl. BERGE, Eléments de combinatoire, Dunod, Paris, 1968, pp.114-117 (Principles of Combinatorics, Academic Press, 1971). O. BERNARDI, N. BONICHON (2007) Catalan's intervals and realizers of triangulations, in Proceedings of the 19th International Conference on Formal Power Series and Algebraic Combinatorics, Nankai University, Tianjin, China (15 pages, http://igm.univmlv.fr/~fpsac/FPSAC07/contrib_papers.html). G. BIRKHOFF, Lattice Theory, American Mathematical Society, Providence, R.I. 1940, 1948, 1967. A. BJÖRNER (1984) Orderings of Coxeter groups, in Combinatorics and Algebra, ed. C. Greene, Contemporary Mathematics 34, American Mathematical Society, Providence, R.I. pp.175-195. A. BJÖRNER, M. L. WACHS (1996) Shellable Nonpure Complexes and Posets, Transactions of the American Mathematical Society 348, 1299-1327. A. BJÖRNER, M. L. WACHS (1997) Shellable non-pure complexes and posets. II Transactions of the American Mathematical Society 397, 3945–3975. N. BONICHON (2005) A bijection between realizers of maximal plane graphs and pairs of non-crossing Dyck paths, Discrete Mathematics 298, 104-114. N. CASPARD (1999) A characterization for all interval doubling schemes of the lattice of
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permutations, Discrete Mathematics & Theoretical Computer Science 3(4), 177–188. N. CASPARD (2000) The lattice of Permutations is bounded, International Journal of Algebra and Computation, 10(4), 481-489. N. CASPARD, C. LE CONTE de POLY-BARBUT, M. MORVAN (2004) Cayley lattices of finite Coxeter groups are bounded, Advances in Applied Mathematics 33(1), 71–94. C. CHAMENI-NEMBUA, Permutoèdre et choix social, thèse de doctorat de mathématiques, Université de Paris V, 1989. C. CHAMENI-NEMBUA (1989) Règle majoritaire et distributivité dans le permutoèdre, Mathématiques, Informatique et Sciences humaines 108, 5-22. C. CHAMENI-NEMBUA, B. MONJARDET (1992) Les treillis pseudocomplémenté finis, European Journal of Combinatorics 13, 89-107. C. CHAMENI-NEMBUA, B. MONJARDET (1993) Finite pseudocomplemented lattices and "permutoèdre", Discrete Mathematics 111, 105-112. F. CHAPOTON (2006) Sur le nombre d'intervalles dans les treillis de Tamari, Séminaire Lotharingien de Combinatoire, 55. J. CRAVEN (1992) Further notes on Craven's conjecture, Social Choice and Welfare 11, 283-285. J. CRAVEN (1996) Majority consistent preference orderings, Social Choice and Welfare 13, 259-267. V. I. DANILOV, A.V. KARZANOV, G. A., KOSHEVOY (2012) Condorcet Domains of tiling type, Discrete Applied Mathematics, to appear. V. I. DANILOV, G.A. KOSHEVOY (2012) Maximal Condorcet Domains, Order, to appear. P. DEHORNOY (2010) On the rotation distance between binary trees, Advances in Mathematics 223, 1316-1355. P. DEHORNOY (2011) Tamari Lattices and the symmetric Thompson monoid, submitted, (http://hal.archives-ouvertes.fr/hal-00626262/fr/). V. DUQUENNE, A. CHERFOUH On the permutation lattice, report CAMS 1991. V. DUQUENNE, A. CHERFOUH (1994) On permutation lattices, Mathematical Social Sciences 27(1), 73-89. B. DUSHNIK, E.W. MILLER (1941) Partially ordered sets, American Journal of Mathematics 63, 600-610. P. H. EDELMAN (1980) Chain enumeration and noncrossing partitions, Discrete Mathematics 31(2), 171–180. P. H. EDELMAN, R. SIMION (1994) Chains in the lattice of noncrossing partitions, Discrete Mathematics 126 (1-3), 107–119. P. H. EDELMAN, C. GREENE (1984) Combinatorial correspondences for Young tableaux, balanced tableaux, and maximal chains in the weak Bruhat order of Sn, in Combinatorics and Algebra, ed. C. Greene, Contemporary Mathematics 34, American Mathematical Society, Providence, R.I. , pp.155-162. P. H. EDELMAN, C. GREENE (1987) Balanced tableaux, Advances in Mathematics 63(1), 42-99. P. H. EDELMAN, R. SIMION (1994) Chains in the lattice of noncrossing partitions, Discrete Mathematics 126 (1-3), 107–119. P.H. EDELMAN, V. REINER (1996) The higher Stasheff-Tamari posets, Mathematika 43, 127-154. P.C. FISHBURN (1992) Notes on Craven's conjecture, Social Choice and Welfare 9, 259262.
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P.C. FISHBURN (1996) Decision theory and discrete mathematics, Discrete Applied Mathematics 68(3), 209-221. P.C. FISHBURN (1997) Acyclic sets of linear orders, Social Choice and Welfare 14, 113124. P.C. FISHBURN (2002) Acyclic sets of linear orders: A progress report, Social Choice and Welfare 19(2), 113-124. P.C. FISHBURN (2005) Subcyclic sets of linear orders, Social Choice and Welfare 24(2), 199-210. S. FLATH (1993) The order dimension of multinomial lattices, Order 10, 201-219 S. FLATH, Multinomial lattices, An approach to the structure of multipermutations via formal concept analysis, Doctoral Dissertation Technischen Hochschule Darmstadt, 1994. L. FREY, Parties distributives du treillis des permutations, in Ordres totaux finis, Mathématiques et Sciences de l'Homme XII, Paris, Gauthier-Villars, pp. 115-126, 1971. L. FREY, M. BARBUT, Techniques ordinales en analyse des données. Algèbre et Combinatoire, Hachette, Paris, 1971. H. FRIEDMAN, D. TAMARI (1967) Une structure de treillis fini induite par une loi demi-associative, Journal of Combinatorial Theory A 2, 215-242. O. FRINK (1962) Pseudo-complements in semi-lattices, Duke Mathematical Journal 29, 505-514. V. GEYER (1994) On Tamari lattices, Discrete Mathematics 133, 99-122. V. GLIVENKO (1929) Sur quelques points de la logique de Brouwer, Bulletin Académie des Sciences de Belgique 15, 183-188. G. Th. GUILBAUD (1952) Les théories de l'intérêt général et le problème logique de l'agrégation, Economie Appliquée 5(4), 1952, reprinted in Éléments de la théorie des jeux, Dunod, Paris, 1968. G. Th. GUILBAUD et P. ROSENSTIEHL (1963) Analyse algébrique d'un scrutin, Mathématiques et Sciences humaines 4, 9-33, and in Ordres totaux finis, Mathématiques et Sciences de l'Homme XII, Paris, Gauthier-Villars, pp. 71-100, 1971. G. GRÄTZER General lattice Theory, Birkhäuser Verlag, Stuttgart, 1978. S. HUANG, D. TAMARI (1972) Problems of associativity: a simple proof for the lattice property of systems ordered by a semi-associative law, Journal of Combinatorial Theory A 13, 7-13. D. HUGUET (1975) La structure du treillis des polyèdres de parenthésage, Algebra Universalis 5, 82-87. C. R. JOHNSON Remarks on mathematical social choice, Working paper 78-25, Dept of Economics, University of Maryland, College Park, 1978. T. KATRINAK (1973) The structure of distributive double p-algebras, regularity and congruences, Algebra Universalis 3, 238-246. D.E. KNUTH "Sorting and searching", The Art of Computer Programming, vol.3, Reading Mass., Addison Wesley, 1973, 2d edition 1998. H. LAKSER (1978) Exercises 30 et 31 p. 51 in G. Grätzer, General lattice Theory, first edition, Birkhäuser Verlag, Stuttgart (p.68 in second edition, 1998). B. LECLERC (1994) A finite Coxeter group whose weak Bruhat order is not symmetric chain, European Journal of Combinatorics 15, 181-185. C. LE CONTE de POLY-BARBUT Personal communications, 1986-1990. C. LE CONTE de POLY-BARBUT (1990) Automorphismes du permutoèdre et votes de Condorcet, Mathématiques, Informatique et Sciences humaines 111, 73-82. C. LE CONTE de POLY-BARBUT (1990) Le diagramme du treillis permutoèdre est intersection des diagrammes de deux produits directs d'ordres totaux, Mathématiques,
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D. TAMARI Monoides préordonnés et chaines de Malcev, Thèse de doctorat, Université de Paris, 1951. D. TAMARI (1962) The algebra of bracketings and their enumeration, Nieuw Archief voor Wiskunde III, 10,131-146. J. VARLET (1963) Congruences dans les treillis pseudo-complémentés, Bulletin de la Société Royale des Sciences de Liège 9, 623-635. J. VARLET (1963) Contributions à l'étude des treillis pseudo-complémentés et des treillis de Stone, Mémoires de la Société Royale des Sciences de Liège 8, 1-71. J. VARLET (1972) A regular variety of type , Algebra Universalis 2, 218223. J. VARLET Structures algébriques ordonnées, Notes d'un séminaire Université de Liège, (134 pages), 1974-75. T. YANAGIMOTO, M. OKAMOTO (1969) Partial orderings of permutations and monotonicity of a rank correlation statistic, Annals of the Institute of Statistical Mathematics 21, 489-506.
