A representation theorem for lattices via set-colored posets

Transcription

1 A representtion theorem for ltties vi set-olore posets Mihel Hi n Lhouri Nourine Astrt This pper proposes representtion theory for ny lttie vi set-olore posets, in the spirit of Birkhoff for istriutive ltties, n Korte n Lovász [1985], Eelmn n Sks [1988] for upper lolly istriutive ltties n onvex geometries. We show tht set-olore posets ptures the orer inue y join-irreuile elements of lttie s Birkhoff s representtion oes for istriutive ltties, i.e. to stuy ltties theory is relte to the oloring of the join-irreuile elements. We lso survey some onsequenes of this representtion on lttie theory n the lttie of Moore fmilies. Keywors: lttie, representtion theory, upper lolly istriutive lttie, losure system, ntimtroi, setolore poset. This pper is motivte y representtion theory n its lgorithmi onsequenes for omintoril ojets struture s ltties. Whenever you re fmilir with Birkhoff s theorem, the intuition ehin this new representtion is the following: Tke poset, sy P = (X, ), set of olors M n olor the elements of P y susets of M. Then the set of olors of ll iels of P hs lttie struture n every lttie n e otine in this wy. For exmple if eh element of P hs only extly one olor then the otine lttie is n ntimtroi. Moreover, if ny pir of elements hve ifferent olors, then the lttie is istriutive. The question " Why evelop lttie theory?" ws onsiere y Birkhoff [5, 6] n extene y Wille [40, 1] using Forml Conept Anlysis (FCA). LIAFA, UMR 7089 CNRS & Université Pris Dierot, F-7505 Pris Ceex 13, Frne LIMOS, UMR 6158 CNRS & Université e Clermont, Cmpus es ézeux F63173 Auière Ceex, Frne 1

2 The purpose of this pper is to present new representtion of ltties tht llow us to unerstn ltties theory from n lgorithmi point of view. Let us first rell the fmous Birkhoff s representtion theorem for istriutive ltties. All results in this pper n e stte for the ul, y repling join-irreuile with meet-irreuile elements. Theorem 1 [3] Any istriutive lttie L is isomorphi to the lttie of ll orer iels I(J(L)), where J(L) is the poset inue y the set of join-irreuile elements of L. Birkhoff s theorem hs een wiely use to erive lgorithms in mny res. In ft, whenever set of ojets hs struture isomorphi to istriutive lttie, then there exists poset where the set of its orer iels is isomorphi to the lttie, e.g. stle mrrige [15], stle llotion [1], minimum uts in network [19], et... For generl ltties, the unique well known representtion is se on sets [4] or inry reltion etween join-irreuile n meet-irreuile elements; lle the iprtite irreuile poset Bip(L) = (J(L), M(L), L ) y Mrkowsky [8, 9, 30] or ontext B(L) = (J(L), M(L), L ) y Wille in the frmework of Forml Conept Anlysis (FCA) [40, 1]. From the FCA perspetive, elements in J(L) re interprete s ojets n those of M(L) re unerstoo s properties or ttriutes hrterizing these ojets. Severl other lttie representtions hve een propose suh s losure systems, implitionl systems, join-ore, ut ll of them use exponentil size (see [10, 8,, 6]). Ltties representtion, FCA n its lgorithmi spets re essentil topis in t nlysis, s they im t ientifying knowleges n restruturing them s hierrhy (the reer is referre for exmples to the ICFCA series of onferenes). Over the pst two ees, mny lgorithms hve een introue to onsier the reonstrution of lttie from its representtion. Fortuntely there exist liner time reonstrution lgorithms for istriutive lttie (see for exmple [17, 0, 39]). Enumertion or reonstrution lgorithms for generl ltties re equivlent to enumerte mximl iliques of iprtite grphs (see [13] for etile nlysis). Compre to the representtion of istriutive ltties, the iprtite irreuile poset oes not tke into ount the orer inue y join-irreuile elements n the ft tht the elements of the lttie orrespon to some orer iels of the poset J(L) = (J(L), ). In this sense the propose lgorithms for the generl se hve ehvior whenever the lttie is istriutive or ner to e istriutive. In this pper, we propose new representtion for generl ltties vi set-olore posets whih generlizes the notion of olore posets for representing upper lolly

3 istriutive lttie in [34, 35]. For lttie L, this representtion ptures the orer inue y join-irreuile elements J(L) s Birkhoff s representtion oes for istriutive ltties. First, we rgue tht this representtion y the ft tht the elements of lttie L orrespon to some orer iels of J(L). When restrite to istriutive ltties, we otin n isomorphism etween L n the orer iels of J(L). Dilworth [9] hs introue upper lolly istriutive ltties n hs oserve tht they re lose to istriutive ltties. Using set-olore posets we onfirm this ie n for upper lolly istriutive ltties, we otin hrteriztion strongly linke to tht of istriutive ltties, whih n e lso eue from Korte n Lovász [5] n Eelmn n Sks [11] works. Reently, Knuer [4, 3] hs onfirme this oservtion using ntihins prtition. Moreover Mgnien et l. [7] hve shown tht onfigurtions of Chip-Firing gme re struture s upper lolly istriutive lttie (see Kolj[4]). The rest of the pper is struture s follows: in Setion 1, we introue the set-olore posets n the lttie of iel olor sets. We lso hrterize set-olore posets whih re ssoite to lttie. In Setion, we list some pplitions of set-olore posets n the lst setion is evote to some lgorithmi onsequenes of our representtion Theorem. 1 Set-Colore Posets In this setion we first introue the notion of set-olore posets n some nottions tht will e use throughout this pper. For efinitions on ltties n orere sets not given here, see [8, 38, 3]. A prtil orer (or poset) on set X is inry reltion on X whih is reflexive, nti-symmetri n trnsitive, enote y P = (X, ). A set I X is si to e n iel if x I n y x implies y I. For n element x X we ssoite the unique iel x = {y X y x}. The set of ll iels of P is enote y I(P ). Let A X, n element z X is n upper oun of A if x z for ny x A. If z is si to e the lest upper oun if z z for ll upper ouns z of A. Dully, we efine the gretest lower oun. A prtil orer L = (X, ) is lle lttie if for every two elements x, y X oth the lest upper oun n the gretest lower oun exist, enote y x y n x y. Let L = (X, ) e lttie. The element z X is join-irreuile (resp. meet-irreuile) if z = x y (resp. z = x y) implies z = x or z = y. The set of ll join-irreuile (resp. meet-irreuile) elements of L is enote y J(L) (resp. M(L)). 3

4 Let L e finite lttie n x, y L. We will use the following rrow reltions [1], tht re wekening of the so lle perspetivities reltions efine in ltties (see [14]) : x y mens tht x is miniml element of {z L z y}, x y mens tht y is mximl element of {z L z x} n x y mens tht x y n x y. Rell tht,, re reltions efine on J(L) M(L). A set-oloring γ for poset P = (X, ) is funtion tht ssigns set of olors to every element in X suh tht for ll x, y X, the sets of olors γ(x) n γ(y) re isjoint whenever x < y. In other wors, γ is set-oloring of the omprility grph of P, s introue in [37]. Definition 1 A set-olore poset, enote y P = (X,, γ, M), is the poset (X, ) equippe with set oloring γ : X M where M is set of olors. A set-olore poset is si to e proper olore poset if the olor set of ny element is singleton, i.e., the oloring γ is funtion from X to M. Figures 1() n 1() show two exmples of set-olore posets f 5 f 5 f e e e () () () Figure 1: () olore poset, () set-olore poset n () the lttie of iel olor sets of the set-olore poset in (). Let P = (X,, γ, M) e set-olore poset. A suset C M is si to e n iel olor set if there exists n iel I of P suh tht C = γ(i) = x I γ(x). In Figure 1(), C = {1, 3, 4, 5} is n iel olor set, sine C = γ({, }). Note tht two ifferent iels of P n hve the sme olor set, i.e. if γ({, e}) = γ({, }) = {1, 3, 4, 5}. The set of ll iel olor sets of P, enote y C(P ) hs lttie struture s shown in the following: 4

5 Proposition 1 Let P = (X, <, γ, M) e set-olore poset. uner set-inlusion is lttie. Then C(P ) orere Proof: It suffies to show tht C(P ) is lose uner union n ontining the empty set. First we show tht C(P ) is lose uner union. Let C 1, C e two iel olors sets of P. Then there exist two iels I 1 n I suh tht γ(i 1 ) = C 1 n γ(i ) = C. Sine iels re lose uner union, thus I 1 I is n iel n therefore C 1 C re its olors. Moreover the iel olor set orresponing to the empty orer iel is empty. Figure 1() shows the lttie of the iel olors sets of the set-olore poset in Figure 1(). Proposition Let P = (X, <, γ, M) e set-olore poset. The mpping gen : C(P ) I(P ) efine y gen(c) = {x X γ( x) C} is n orer emeing. Moreover, gen(c) is the unique lrgest iel I of P with γ(i) = C. Proof: Let C 1, C e two iel olors sets of P. We show tht C 1 C iff gen(c 1 ) gen(c ). First suppose C 1 C n let x gen(c 1 ). Then γ( x) C 1 C whih implies tht x gen(c ). Now suppose tht C 1 C. Then there exists C 1 \ C n x X suh tht γ(x), whih implies tht γ(x) C n therefore gen(c 1 ) gen(c ) sine x / gen(c ). Assume tht there exist two ifferent mximl (uner inlusion) iels I n J with γ(i) = γ(j) = C. Then γ(i J) = C sine iels re lose uner union, n thus ontrits the ft tht I n J re mximl uner inlusion with γ(i) = γ(j) = C. Let us now exmine the onsequenes of these efinitions. Representing lttie y set-olore poset In this setion we show tht ny lttie L n e represente y set-olore poset P L suh tht its ssoite lttie C(P L ) is isomorphi to L. Definition Let L = (X, ) e lttie. We enote P L = (J(L),, γ, M(L)), the set-olore poset efine y the following set-oloring : γ : J(L) M(L), with γ(j) = {m M(L) j m} 5

6 () 34 () 1 Figure : () lttie L in whih join-irreuile (resp. meet-irreuile) elements re lelle with letters (resp. numers) n () its ssoite set-olore poset P L. Lemm 1 Let L = (X, ) e lttie n the mpping ϕ : L C(P L ) with ϕ() = γ(j()), where J() = {j J(L) j }, then: 1. For every X, ϕ() = {m M(L) m}. For every, X, iff ϕ() ϕ(), i.e. ϕ is n orer emeing. Proof: 1. Let m M(L) n m. Then there exists j suh tht j m. Thus m γ(j) whih implies m γ(j()) = ϕ(), sine j J(). Now let m ϕ(). Then there exists j J() suh tht j m. This mens tht j m n then m sine j.. Let, for, L. Then m implies m. So ϕ() ϕ(). Now let ϕ() ϕ(). This mens, tht for ll m M(L), m implies m. Suppose. Then there exists m M(L) with m et m, whih is ontrition. We n now formulte our min representtion theorem : 6

7 Theorem Any lttie L is isomorphi to the lttie of olore iels of its poset P L. Proof: To this im, let us prove tht for lttie L = (X, ), ϕ : L C(P L ) is n orer-isomorphism, with ϕ() = γ(j()) n ϕ 1 (C) = gen(c). Using Lemm 1, the mpping ϕ is n orer emeing n therefore one-to-one. It remins to show tht ϕ is onto. Let C C(P L ) n = gen(c). Then L sine the supremum lwys exists for finite lttie. It suffies to show tht ϕ() = C. Let m C. Then there exists j suh tht m γ(j), i.e. j m. Thus m n m ϕ(). Conversely, let m ϕ(). Then there exists j suh tht j m. Suppose m C. This implies tht j m for ll j gen(c). Therefore = gen(c) m whih ontrits m n then m ϕ(). As Birkhoff s Theorem 1 whih provies for istriutive ltties not only ompt representtion vi poset ut lso some struturl insights tht n e use lgorithmilly [17], our Theorem oes the sme for ritrry ltties. This representtion is ompt sine L n e exponentil in P L. Before isussing some of the onsequenes of this result, let us first hrterize set-olore posets whih re isomorphi to the set-olore poset P L for some lttie L. Figure 3 shows three set-oloring of the sme poset n none of them is isomorphi to some P L for lttie L. Rell the hrteriztion of those iprtite posets whih re isomorphi to Bip(L) = (J(L), M(L), ) for some lttie L. Theorem 3 [30] Let B = (X, Y, E) e iprtite poset. B is isomorphi to Bip(L) = (J(L), M(L), ) for some lttie L if n only if the following hols: 1. For ll x X, if W X is suh tht N(x) = N(W ), then x W.. For ll y Y, if V Y is suh tht N(y) = N(V ), then y V. where N(x) is the neighoorhoo of x in B. Using the hrteriztion of Theorem 3, we erive hrteriztion of setolore posets whih re isomorphi to P L for some lttie L. Theorem 4 Let Q = (X,, γ, M) e set-olore poset. Then Q is isomorphi to P L for some lttie L iff the following onitions re stisfie : 7

8 3 4 3 e e e ) ) ) () () () e e e () (e) (f) Figure 3: (), (e) n (f) re respetively the ltties of iel olor sets of set-olore posets in (),() n (). In () the element is not join-irreuile, in () the element is not omprle to e n in () the numer of olors is greter thn the numer of meet-irreuile elements. 1. For ll A X n x X, γ( x) = γ( A) implies x A (irreuile onition), where x = {y X y x} n γ( A) = A γ().. For ll x, y X, γ( x) γ( y) implies x y (orering onition). 3. For ll m, n, p M, β(m) β(n) or β(m) = β(n) β(p) implies m = n or m = p, where β(m) = { X m γ( )}. Proof: These onitions re oviously neessry, let us exmine their suffiieny. Using irreuile onition n Lemm 1, there is ijetion etween X n J(L), n y () (X ) is isomorphi to (J(L), ). Similrly there is ijetion etween 8

9 Y n M(L), tht ssoite to m Y the iel olor set C = γ({ m γ( )}). Now, let m γ(), X. Then γ( ) C sine m C. Clerly onitions of Theorem 4 n e heke in polynomil time in the size of the olore poset n thus it n e reognize in polynomil time wether given set-olore poset is isomorphi to some P L for lttie L. 3 Applitions on prtiulr lsses of ltties We show in this setion how Theorem unifies mny results of lttie theory. First, we erive the fmous Birkhoff s representtion for istriutive ltties Theorem 1, then we onsier Korte n Lovász s results [5] or Eelmn n Sks s results [11] for upper lolly istriutive ltties. We lso show tht Theorem yiels hrteriztion of the lttie of ll Moore fmilies n lso of extreml n semiistriutive ltties. 3.1 Distriutive ltties Distriutive ltties hve use either theoretilly or lgorithmilly in severl res. In ft severl omintoril ojets n e struture s istriutive ltties (see for exmple, Knuer [4] who gives list of prolems from grphs). Theorem 5 Let L e lttie. Then the following re equivlents: 1. L is istriutive. L is isomorphi to the lttie of ll iels of the poset inue y join-irreuile elements of L 3. For eh j J(L) there exists unique m M(L) suh tht j m n ully, for eh m M(L) there exists unique j J(L) suh tht j m. 4. P L = (J(L),, γ, M) is proper olore poset n γ is ijetive. Proof: The equivlene etween 1. n. is ue to Birkhoff s Theorem 1. The equivlene etween 1. n 3. is ue to Theorem of Wille [1]. To show tht. is equivlent to 4., it suffies to note tht there is ijetion etween orer iels of (J(L), ) n the iel olor sets of P L. Inee, two ifferent iels hve ifferent olors sine γ is ijetive. The equivlene etween 3. n 4. is y efinition of P L. 9

10 Figure 4: The istriutive se 3. Upper lolly istriutive ltties Dilworth [9] hs introue upper lolly istriutive ltties n hs oserve tht they were lose to istriutive ltties. Upper lolly istriutive ltties hve een reisovere mny times, n hve severl nmes in the literture (suh s joinistriutive ltties [11], or ntimtrois [5],... ). For survey, see Monjret [31] whih ontins mny hrteriztions. Here we will onsier n upper lolly istriutive lttie s n ntimtroi. Bse on our work [34], Knuer [4] hs given n equivlent hrteriztion n gives list of pplitions of upper lolly istriutive ltties. Using the hrteriztion of upper lolly istriutive lttie y rrows reltions, we otin the following: Figure 5: An exmple of lolly istriutive lttie n its ssoite proper olore poset Theorem 6 [34] For lttie L, the following sttements re equivlent: 10

11 1. L is upper lolly istriutive.. For eh j J(L) there exists unique m M(L) suh tht j m. 3. P L = (J(L),, γ, M(L)) is proper olore poset. Proof: The equivlene etween 1. n. is ue to Gnter n Wille [1] (ul of Theorem 44). The equivlene etween 3. n 4. is y efinition of P L. Therefore, istriutive n upper lolly istriutive ltties n e represente y proper olore posets. Furthermore, Theorem 6 onfirms Dilworth s oservtion, sine it iffers only slightly from Theorem The lttie of Moore fmilies In this setion we rell the hrteriztion of the lttie of Moore fmilies using proper olore poset [16]. Let X e n n-set n X its power set. A Moore fmily on X is fmily of susets of X lose uner set-intersetion n ontining the set X. A Moore fmily is lso known s losure system; i.e. the set of ll lose sets of losure opertor. The set M n of ll possile Moore fmilies on n n-set X, orere y set-inlusion is lower lolly istriutive lttie. By Theorem 6 there exists olore poset P n suh tht the lttie M n is ully isomorphi to the set of its iel olor sets. Let Q e oolen lttie on n toms, sy 0, 1,..., n 1. We onsier the mpping γ : Q [0, n 1] s follows : 0 if x is the ottom element γ(x) = i if x = i for some i [0, n 1] (eq 1) J(x) γ() otherwise where J(x) is the set of ll toms elow x in Q. The pplition γ is properly oloring sine eh element of X hs only one olor. Moreover x < Q y implies J(x) J(y) n therefore γ(x) γ(y). The poset P n is efine s the isjoint sum of ll intervls [, ] of Q where is n tom of Q n the top element of Q, i.e. P n = n 1 0 Q i where Q i is the inue poset y [ i, ] in Q n the oloring is inherite from the oloring of Q (see Figure 6). Proposition 3 [16] There is ijetion etween the iel olors sets of P n n the set of Moore fmilies on n n-set. 11

12 {1,,3} 3 {1,3} {,3} {1,} {1} {} 0 ) 1 ) {} ) {},{1},{},{1,} {},{},{1,} {},{1,} {},{1},{1,} {},{} {},{1} ) {} Figure 6: ) the oolen lttie hving two toms, ) P the properly olore poset, ) the iel olor sets lttie of P, n ) the lttie of Moore fmilies on the set X = {1, }. By pplying the lgorithm to generte iel olor sets for olore poset, we foun tht the numer of Moore fmilies on 6 elements is extly (see [16] for more etils). Reently, Colom et l. [7] hve otine the numer of Moore fmilies on n = 7 using reursive eomposition of M n. Now we erive n expliit hrteriztion the lttie of ll Moore fmilies using the inry reltion R n = (J n, M n, ) where J n n M n re respetively the set of joinirreuile n meet-irreuile elements. This is rewriting of Proposition 3 using inry reltion inste of olore poset. Let J n = { ij suh tht i [0, n 1], j [0, n 1 1]}, M n = {k [1, n 1]} n ij k iff ( i OR j ) AND k = k where OR n AND re the logil inry opertions. Corollry 1 There is ijetion etween mximl ntihins (iliques ) of R n = (J n, M n, ) ( R n = (J n, M n, )) n the set of ll Moore fmilies on n-set. 3.4 Semiistriutivity n extremlity In [18] meet-simpliil ltties were introue. This lss of ltties generlizes mny known lsses of ltties suh s meet-extreml n meet-semiistriutive ltties. Using set-olore posets the uthors in [18] hve hrterize meet-semiistriutive ltties s Ntion [33] i for semiistriutive ltties. 1

13 A lttie L is si to e meet-semiistriutive if for ll elements x, y, z L, x y = x z implies x y = x (y z). A meet-semiistriutive lttie is si semiistriutive if for ll elements x, y, z, x y = x z implies x y = x (y z). L is lle meetextreml if for ll x L h(l) = M(L) where h(l) is the size of mximl hin in L. A meet-extreml lttie L is lle extreml if h(l) = J(L). Let L e lttie with n join-irreuiles n σ = j 1, j,..., j n e n orering of J(L). We enote y i the olors tht not pper in the i-1 first join-irreuiles in P L, ie. i = γ(j i ) \ i 1 h=1 γ(j h). L is lle meet-simpliil if there is totl orering σ = (j 1,..., j J(L) ) suh tht i 1. The following results on extreml n meet-extreml theorems re written in terms of set-olore posets [30]. Theorem 7 [18] A lttie L is semiistriutive (resp. extreml) then there exists n orering (resp. liner extension) σ = j 1, j,..., j n of J(L) suh tht i = 1. Theorem 8 [18] If L is meet-extreml (resp. meet-semiistriutive) lttie then there exists n orering (resp. liner extension) σ = j 1, j,..., j n of J(L) suh tht i 1. We onlue tht meet-extreml n meet-semiistriutive istriutive ltties re meet-simpliil. 4 Disussion n opens prolems 1. In orer to formlize the proximity to istriutive ltties, let us efine new lttie invrint lle hromti inex : For ny ltttie L n its ssoite olore poset P = (X, <, γ, M), we efine χ(l) s mx x X γ(x). We hve notie tht for upper lolly istriutive ltties the hromti inex is one. A nturl prolem rises: Fin hrteriztion of ltties hving hromti inex k, for every k? Reently, Beuou et l [] hve shown tht omputing miniml implitionl sis from set-olore poset n the opposite n e one in polynomil time whenever the hromti inex is onstnt.. Cn we list the set of ll iel olors sets in O(n ) per element using polynomil spe, where n is the numer of verties of J(L)? The est known omplexity is relte to the mtries prout, i.e. O(n.38 ). Nourine n Rynu [36] hve given n O(n ) lgorithm ut using exponentil spe. 13

14 3. Cn we ompute for lttie given y its set-olore poset, miniml implitionl sis in qusi-polynomil? The unique qusi-polynomil time lgorithm for ltties orrespons to ltties isomorphi to n inepenene system (or hypergrph), where miniml sis is the set of ll miniml trnsversl of the hypergrph. Reently, Knte et. l [1] hve shown tht the enumertion of miniml trnsversl of n hypergrph is polynomilly equivlent to the enumertion of miniml ominting sets of grph. Connete miniml ominting sets n e nite for the generl se? We hope tht this representtion of ltties using set-olore posets oul e helpful for stuying lttie theory n its lgorithmi spets using the ft tht setolore posets is simple generliztion of Birkhoff s representtion for istriutive ltties. Referenes [1] Mour Bïou n Mihel Blinski. The stle llotion (or orinl trnsporttion) prolem. Mth. Oper. Res., 7(4):66 680, Novemer 00. [] Lurent Beuou, Arnu Mry, n Lhouri Nourine. Algorithms for k-meetsemiistriutive lttie. Sumitte, 015. [3] G. Birkhoff. Lttie Theory, volume 5 of Coll. Pul. XXV. Amerin Mthemtil Soiety, Proviene, 3r eition, [4] G. Birkhoff n O. Frink. Representtions of ltties y sets. Trns. Amer. Mth. So., [5] Grret Birkhoff. Ltties n their pplitions. Bull. Amer. Mth. So, 44: , [6] Grret Birkhoff. Wht n lttie o for you? In In Trens in Lttie Theory (J.C. Aot, e.), pges Vn Nostrn-Reinhol, New York, [7] Pierre Colom, Alexis Irlne, n Olivier Rynu. Counting of moore fmilies for n=7. In ICFCA, pges 7 87, 010. [8] B. A. Dvey n H. A. Priestley. Introution to ltties n orers. Cmrige University Press, seon eition,

15 [9] R. P. Dilworth. Ltties with unique irreuile eompositions. Ann. Of Mth., (41): , [10] V. Duquenne. The ore of finite lttie. Disrete Mth., 88: , [11] P. H. Eelmn n M. E. Sks. Comintoril representtion n onvex imension of onvex geometries. Orer, [1] B. Gnter n R. Wille. Forml Conept Anlysis: Mthemtil Fountions. Springer-Verlg Berlin, [13] Alin Gély, Lhouri Nourine, n Bhir Si. Enumertion spets of mximl liques n iliques. Disrete Applie Mthemtis, 157(7): , 009. [14] G. Grätzer. Generl Lttie Theory. Birkhäuser, [15] Dn Gusfiel, Roert Irving, Pul Lether, n Mihel Sks. Every finite istriutive lttie is set of stle mthings for smll stle mrrige instne. Journl of Comintoril Theory, Series A, 44(): , [16] M. Hi n L. Nourine. The numer of moore fmilies on =6. Disrete Mthemtis, 94(3):91 96, 005. [17] Mihel Hi, Roul Mein, Lhouri Nourine, n George Steiner. Effiient lgorithms on istriutive ltties. Disrete Applie Mthemtis, 110(-3): , 001. [18] P. Jnssen n L. Nourine. A simpliil elimintion sheme for meetsemiistriutive ltties n intervl ollpsing. Alger Universlis, 003. [19] Pir Jen-Clue n Queyrnne Murie. On the struture of ll minimum uts in network n pplitions. In V.J. Rywr-Smith, eitor, Comintoril Optimiztion II, volume 13 of Mthemtil Progrmming Stuies, pges [0] R. Jégou, R. Mein, n L. Nourine. Liner spe lgorithm for on-line etetion of glol preites. pges Strutures in Conurreny Theory, Berlin 1995, J. Desel e. Workshops in Computing, Springer, [1] Mmou Moustph Knté, Vinent Limouzy, Arnu Mry, n Lhouri Nourine. On the enumertion of miniml ominting sets n relte notions. SIAM J. Disrete Mth., 8(4): ,

16 [] D. Kelly. Comprility grphs. In I. Rivl, eitor, Grphs n Orers, pges D. Reiel Pulishing Compny, [3] Kolj B. Knuer. Chip-firing, ntimtrois, n polyher. Eletroni Notes in Disrete Mthemtis, 34:9 13, 009. [4] Kolj B. Knuer. Ltties n Polyher from grphs. PhD thesis, Tehnishen universitt Berlin, 010. [5] B. Korte n L. Lovász. Homomorphisms n rmsey properties of ntimtrois. Dis. Appl. Mth, 15:83 90, [6] H.M. MNeille. Prtilly orere sets. Trns. of the Amer. Mth. So., 4:90 96, [7] Clémene Mgnien, H Duong Phn, n Lurent Vuillon. Chrteriztion of ltties inue y (extene) hip firing gmes. In DM-CCG, pges 9 44, 001. [8] G. Mrkowsky. Comintoril spets of lttie theory with pplitions to the enumertion of free istriutive ltties. PhD thesis, Hrvr University, Cmrige, Msshsetts, [9] G. Mrkowsky. Some omintoril spets of lttie theory. In Houston Lttie Theory Conf., eitor, Pro. Univ. of Houston, pges 36 68, [30] G. Mrkowsky. Primes, irreuiles n extreml ltties. Orer, 9:65 90, 199. [31] B. Monjret. The onsequenes of ilworth s work on ltties with irreuile eompositions. In R. Freese K.P. Bogr n J. Kung, eitors, The Dilworth theorems selete ppers of Roert P. Dilworth, pges Birkhäuser, [3] B. Leler N. Cspr n B. Monjret. Finite Orere Sets Conepts, Results n Uses. Cmrige University Press, 01. [33] J. B. Ntion. Unoune semiistriutive ltties. Alger n Logi, 000. [34] L. Nourine. Colore posets n upper lolly istriutive ltties. Tehnil Report RR LIRMM 00060, Université Montpellier II, LIRMM,

17 [35] L. Nourine. Une Struturtion Algorithmique e l Théorie es Treillis. Hilittion à iriger es reherhes, Université e Montpellier II, Frne, July 000. [36] L. Nourine n O. Rynu. A fst lgorithm for uiling ltties. IPL, [37] F.S. Roerts. T-olourings of grphs : Reent results n open prolems. Disrete Mthemtis, [38] B. S. W. Shröer. Orere sets, n introution. Birkhäuser, 00. [39] M. B. Squire. Enumerting the iels of poset. Preprint. North Crolin Stte University, [40] R. Wille. Restruturing lttie theory: An pproh se on hierrhies of ontexts. in Orere sets,i. Rivl, Es. NATO ASI No 83, Reiel, Doreht, Holln, pges ,