A Mathematica package to cope with partially ordered sets

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1 A Mthemtic pckge to cope with ptill odeed sets P. Cod Diptimento di Infomtic e Comunicione, Univesità degli Studi di Milno Abstct Mthemtic offes, b w of the pckge Combintoics, mn useful functions to wok on gphs nd odeed stuctues, but none of these functions ws specific enough to meet the needs of ou esech goup. Moeove, the eisting functions e not lws helpful when one hs to wok on new concepts. In this ppe we pesent pckge of fetues developed in Mthemtic which we conside pticull useful fo the stud of cetin ctegoies of ptill odeed sets. Among the fetues offeed, the pckge includes: () some bsic fetues to tet ptill odeed sets; (2) the bilit to enumete, cete, nd displ monotone nd egul ptitions of ptill odeed sets; (3) the cpbilit of constucting the lttices of ptitions of poset, nd of doing some useful computtions on these stuctues; (4) the possibilit of computing poducts nd copoducts in the ctego of ptill odeed sets nd monotone mps; (5) the possibilit of computing poducts nd copoducts in the ctego of foests (disjoint union of tees) nd open mps (cf. [DM06] fo the poduct between foests).. Intoduction In ou esech we often del with combintoil poblems, sometimes quite comple, on odeed stuctues. Recll tht ptill odeed set (poset, fo shot) is set togethe with efleive, tnsitive, nd ntismmetic bin eltion (usull denoted b b) on the elements of the set. Ou ttention is minl ddessed to cetin ctegoies of posets which hve stong connections with logic, especill with mn-vlued (o fu) logic.

2 2 Cod_UGM200.nb In [Cod08] nd [Cod09], the utho investigtes the notion of ptition of poset. In the clssicl theo, ptition of set S is set of nonempt, piwise disjoint subsets of S, clled blocks, whose union is S. In the poset cse, thee diffeent notions of ptition e given: monotone, egul, nd open. As shown in both woks, the "ight" notion depends on the ctego we e deling with, tht is, depends on the kind of posets nd mps between them tht we e consideing. Essentill, ptition (of n kind) of poset P is poset whose elements e blocks of (set) ptition of the undeling set of P, endowed with n ppopite ptil ode. We do not give hee the foml definition of the thee kinds of ptition, nd we efe the inteested ede to [Cod09]. In [Cod08], it is lso shown tht the sets of ll monotone nd egul ptitions of poset cn be endowed with lttice stuctues, i.e., the e poset such tht n two elements hve supemum nd n infimum. Mn inteesting combintoil questions ise fom these two woks. Fo instnce: how mn ptitions (of the thee kinds) of poset thee e?, e thee some eltions between the thee notions, nd with the clssicl notion of ptition?, cn we give simple, intuitive notion of ptition of poset?, which popeties the lttices of ptitions do hve? The Mthemtic pckge pesented in this ppe ws imed to help nswe to these questions, nd hs been then developed in ode to solve moe genel poblems on posets. In Section 2 we pesent some bsic fetues of ou pckge poset.m. Some of these fetues e just specilition of some functions vible vi the pckge Combintoic. In Section 3 we pesent some specil functions of the pckge, devoted to the genetion, enumetion nd investigtion of ptitions of posets. In Section 4 we ppoch the poblem of ceting the lttice of ptitions of poset, nd t to descibe how to use some fetues of poset.m to find some popeties of the lttice we hve obtined. In Section 5 we pesent two cse studies (tken fom [Cod08]) to show how to use the pckge in two diffeent concete emples. Among the fetues vilble in the pckge, we hve mentioned the possibilit of computing poducts nd copoducts (ctegoicl sum) between posets. In [DM06] the uthos pesent method to compute copoducts of finitel pesented Gödel lgeb (pticul lgebs stictl elted to Gödel logic, mn-vlued logic). Such method is bsed on computing poduct between foests, i.e, posets such tht the unde set of ech element is totll odeed set ( chin). The pckge poset.m include n lgoithm to compute this kind of poduct, besides of couse othe functions to compute Ctesin poducts, nd copoducts. In the finl Section 6 we show how to use the fetues just bove mentioned. 2. Bsic fetues We use the usul commnd to lod the pckge. << poset.m Fist, we wnt to cete poset. A poset is intenll epesented s Gph object. The function Poset llows to cete poset.? Poset Poset@eltionD nd Poset@eltionD,vD genete ptill odeed set, epesented s diected gph Hsee Combintoic mnull. eltion is set of pis epesenting the ode eltion of the poset Hnot necessil the entie ode eltionl. v is list of vetices. B2 = Poset@88"", ""<, 8"", ""<<D Gph:< 5,3,Diected >

3 Cod_UGM200.nb 3 To displ the poset we use the function Hsse. Hsse@B2D The pckge povide functions to llow n es gention of some common posets. C3 = Poset@88c, c2<, 8c2, c3<<d Gph:< 6,3,Diected > CU3 = Chin@3D Gph:< 6,3,Diected > Hsse@8C3, CU3<D c3 3 c2 2 c Clling the function Poset[eltion, v] llows the cetion of posets with isolted points. It is lso possible to cete empt posets. F = Poset@88"", ""<<, 8"0", "", ""<D Gph:< 4,3,Diected > Hsse@FD 0 The pckge povide functions to get diffeent epesenttion of poset, nd to investigte its stuctue (elements, ode eltion,...).? Reltion Reltion@pD gives the list of pis of the ode eltion of the ptill odeed set p. p is Poset, o list of Posets.

4 4 Cod_UGM200.nb 88c,c <, 8c,c 2 <, 8c,c 3 <, 8c 2,c 2 <, 8c 2,c 3 <, 8c 3,c 3 <<? Coveing Coveing@pD gives the list of pis of the coveing eltion of the ptill odeed set p. p is Poset, o list of Posets. Coveing@C3D 88c,c 2 <, 8c 2,c 3 << PosetElements@C3D 8c,c 2,c 3 < 3. Investigting ptitions of posets In this section we show how to wok on ptitions of posets using poset.m. At the moment, the pckge llows to cete nd mnge monotone nd egul ptitions.? PosetPtitions PosetPtitions@pD genetes the list of ll monotone ptitions of poset. Ech monotone ptition is epesented s gph. p is poset. PosetPtitions@pD outputs list of gphs, nd displs the totl numbe of monotone ptition of p, nd the totl numbe of cses nled b the function to obtin the monotone ptitions. PB2 = PosetPtitions@B2D; Anled peodes: 6 Poset Ptitions: 7 The function CeteHsse llows to displ the monotone ptitions s posets. In the following Hsse digms, the lbels of the elements epesent blocks of the ptitions. The e obtined b conctenting the lbels of the elements of ech block in the oiginl poset. CeteHsse@PB2, 3D

5 Cod_UGM200.nb 5 In genel, monotone ptitition etuned b the function PosetPtitions is not poset, but peode (i.e. the bin eltion does not hve the ntismmetic popet). If we ppl the function Hsse to one of such ptitions, we do not obtin n Hsse digm, but diected gph. The function PtitionToPoset solves this poblem, b educing blocks to single elements nd conctenting lbels s mentioned bove. Hsse@PB2@@4DDD? PtitionToPoset PtitionToPoset@pD genetes the poset coesponding to peode seen s ptition. Wheneve two elements nd b e such tht H,bL nd Hb,L belong to the ode eltion of p, the e elements of the sme block, nd the e tnsfomed in single element of new poset. p is gph, o list of gphs. Hsse@PtitionToPoset@PB2@@4DDDD The nlogous of PosetPtition fo geneting egul ptitions of posets is the function RegulPtitions.? RegulPtitions RegulPtitions@pD genetes the list of ll egul ptitions of poset. Ech egul ptition is epesented s gph. p is poset. RegulPtitions@pD outputs list of gphs, nd displs the totl numbe of egul ptition of p, nd the totl numbe of cses nled b the function to obtin the egul ptitions. RB2 = RegulPtitions@B2D; Anled peodes: 5 Regul Ptitions: 5

6 6 Cod_UGM200.nb 3D 4. Investigting the lttices of ptitions As fo set ptitions, monotone (egul) ptitions of poset cn be endowed with lttice stuctue. This cn be done b odeing the monotone (egul) ptitions of the poset b inclusion between the peode eltions epesenting ech ptition (cf. [Cod08, 4]). In the following we see how to constuct nd mnge the monotone nd egul ptition lttices of poset with the pckge poset.m. Some dditionl fetues e lso povided. Fo instnce, we show the use of two functions, Upsets nd Downsets, llowing to obtin the uppe set (o upset) nd the lowe set (o downset) of ech ptition in the lttice. These functions outputs lists of positions, ech of which gives the positions of the ptitions in the input list. The input list is usull obtined b the functions PosetPtitions o RegulPtitions descibed in the pevious section.? Upsets Upsets@plistD etuns the list of the upsets of ech element of plist in the lttice of ptition. plist is list of monotone o egul ptitions, nd cn be obtined b the functions RegulPtitions o PosetPtitions. Upsets@RB2D 88, 2, 3, 4, 5<, 82, 5<, 83, 5<, 84, 5<, 85<< Downsets@RB2D 88<, 8, 2<, 8, 3<, 8, 4<, 8, 2, 3, 4, 5<<? PosetPtitionLttice PosetPtitionLttice@plistD etuns the lttice stuctue of given list of monotone o egul ptitions plist. plist is usull obtined b using RegulPtitions o PosetPtitions. LRB2 = PosetPtitionLttice@RB2D Gph:< 2,5,Diected > The lbels of the elements in the following Hsse digm indicte the position of the egul ptitions in the list RB2 (see the pevious section fo the Hsse digms of the elements of RB2).

7 Cod_UGM200.nb 7 Hsse@LRB2D Hsse@LPB2 = PosetPtitionLttice@PB2DD Some functions povided b the pckge cn help in investigting the popeties of monotone nd egul ptitions lttices.? Moebius Moebius@plistD etuns the vlues of the Moebius function computed on ech element of monotone o egul ptition lttice. plist is list of ptitions usull obtined b using the functions PosetPtitions o RegulPtitions. Moebius@PB2D 8,,, 0,, 0, 0< Atoms nd cotoms e the elements of the lttice tht lie just bove the bottom element, o below the top element, espectivel.? AtomsPosition AtomsPosition@plistD etuns the list of positions of toms of the monotone o egul ptition lttice given in input. plist is list of ptitions usull obtined b using the functions PosetPtitions o RegulPtitions. AtomsPosition@PB2D 82, 3< CotomsPosition@PB2D 84, 5, 6< Some elements of the monotone ptition lttice of poset e isomophic to line etensions of the poset.

8 8 Cod_UGM200.nb Some lttices (nmel, the nked lttices) cn be endowed with nk function such tht HL < HL wheneve < nd such tht wheneve coves, then HL = HL +. The vlue of the nk function fo n element of the lttice is clled its nk. Whitne numbes count the sies of ech level of nked lttice, tht is, the n th Whitne numbe counts the numbe of elements of lttice hving nk n. We ssume tht the bottom element hs nk. As shown in [Cod08], the lttice of egul ptitions of poset is lws nked, while the lttice of poset ptition, in genel, is not.? WhitneNumbes WhitneNumbes@plistD etuns the Whitne Numbes of nked lttice. plist is list of posets foming nked lttice. WhitneNumbes@PB2D 8, 2, 3, <? WhitneLevels WhitneLevels@plistD etuns the high of the elements of lttice. plist is list of posets foming nked lttice. WhitneLevels@PB2D 8, 2, 2, 3, 3, 3, 4< The ltte functions e pticull useful when the gphicl output cnnot offe n infomtion on the lttice. Hsse@P4 = Poset@88"", ""<, 8"", "b"<, 8"b", ""<, 8"", ""<, 8"", "c"<, 8"", "d"<, 8"c", ""<, 8"d", ""<<DD b c d BigLttice = RegulPtitions@P4D; Anled peodes: 877 Regul Ptitions: 49

9 Cod_UGM200.nb 9 Hsse@PosetPtitionLttice@BigLtticeDD WhitneNumbes@BigLtticeD 8, 9, 07, 208, 3, 24, < Atoms = AtomsPosition@BigLtticeD 82, 3, 4, 5, 6, 7, 5, 6, 20, 2, 23, 24, 25, 26, 33, 49, 50, 5, 6<

10 0 Cod_UGM200.nb 4D b b c d c d b c d b cd d c b c b d c b d d b c bc d bd c b c d c d b b d b c b c d c b d c d d b b c b c d c d Cotoms = CotomsPosition@BigLtticeD 8457, 458, 459, 460, 46, 462, 463, 464, 465, 466, 467, 468, 479, 480, 48, 482, 483, 484, 485, 486, 487, 488, 489, 490<

11 Cod_UGM200.nb 7D d cd cd c bc bd b bc b b bd d c cd b b c d d cd cd bd bc bcd bcd bc c bcd b bc bd cd bd cd d c b bcd bcd bcd bcd 5. Cse studies ü 5. Ptitions of chins In [Cod08, 6.2] it is poved tht the monotone ptition lttice nd the egul ptition lttice of chin with n elements e isomophic, nd tht the e isomophic to the Boolen lttice B n-. In this section, we show some smll emple of this fct. Hsse@8PosetPtitionLttice@PosetPtitions@Chin@2DDD, PosetPtitionLttice@PosetPtitions@Chin@3DDD, PosetPtitionLttice@PosetPtitions@Chin@4DDD<D Anled peodes: 2 Poset Ptitions: 2 Anled peodes: 8 Poset Ptitions: 4 Anled peodes: 64 Poset Ptitions: Hsse@8PosetPtitionLttice@RegulPtitions@Chin@2DDD, PosetPtitionLttice@RegulPtitions@Chin@3DDD, PosetPtitionLttice@RegulPtitions@Chin@4DDD<D

12 2 Cod_UGM200.nb Anled peodes: 2 Regul Ptitions: 2 Anled peodes: 5 Regul Ptitions: 4 Anled peodes: 5 Regul Ptitions: Hsse@8BoolenAlgeb@D, BoolenAlgeb@2D, BoolenAlgeb@3D<D ü 5.2 A cse of counting We pesent in this section n enumeting poblem solved in [Cod08, 6.4]. We wnt to count ll egul ptitions of fmil of posets M, M 2, M 3,..., shown below. M = Poset@88"", ""<, 8"", "t"<<d; M2 = Poset@88"", ""<, 8"", "b"<, 8"b", "t"<, 8"", "t"<<d; M3 = Poset@88"", ""<, 8"", "b"<, 8"", "c"<, 8"c", "t"<, 8"b", "t"<, 8"", "t"<<d; M4 = Poset@88"", ""<, 8"", "b"<, 8"", "c"<, 8"", "d"<, 8"d", "t"<, 8"c", "t"<, 8"b", "t"<, 8"", "t"<<d; M5 = Poset@88"", ""<, 8"", "b"<, 8"", "c"<, 8"", "d"<, 8"", "e"<, 8"e", "t"<, 8"d", "t"<, 8"c", "t"<, 8"b", "t"<, 8"", "t"<<d;

13 Cod_UGM200.nb 3 Hsse@8M, M2, M3, M4, M5<, 3D t t t b b c t b c d t b c d e RegulPtitions@ D & ê@ 8M, M2, M3, M4, M5<; Anled peodes: 5 Regul Ptitions: 4 Anled peodes: 5 Regul Ptitions: Anled peodes: 52 Regul Ptitions: 38 Anled peodes: 203 Regul Ptitions: 52 Anled peodes: 877 Regul Ptitions: 675 The following fomul count the totl numbe of egul ptitions of the poset M i. B i+2 B i+ + The numbe B n is the n th Bell numbe, nd it is computed b the Mthemtic function BellB[n]. Tble@BellB@n + 2D BellB@n + D +, 8n,, 5<D 84,, 38, 52, 675, 3264, 7 008, , , , , , , , < 6. Computing poducts nd copoducts We pesent hee some emples of computtion of poducts nd copoducts of posets. Poducts nd copoducts e diffeent depending on the ctego we e woking on. We do not go into detils. Fo the pupose of this wok it is sufficient to s tht if we conside geneic posets s objects (nd if mps between posets e ode peseving) the copoduct woks s disjoint union, nd the poduct woks s Ctesin poduct. If, on the othe hnd, we conside foests s objects (nd pticul mps between them clled open mps) the copoduct is disjoint union nd the poduct cn be computed s descibed in [DM06]. To distinguish the Mthemtic functions fo poduct nd copoducts, thei nmes begin with Poset if we e woking in the ctego of posets, nd the begin with Foest if we e woking in the ctego of foests. F = Poset@88, 2<<D; F2 = Poset@88, <, 82, 3<<D;

14 4 Cod_UGM200.nb 2 3 2? PosetSum computes the sum HcopoductL of the posets p, p2,... in the ctego of posets nd ode peseving mps. The output is poset. F2DD ? FoestSum FoestSum@p,p2,..D computes the sum HcopoductL of the foests p, p2,... in the ctego of foests nd open mps. The output is foest. Hsse@FoestSum@F, F2DD ? PosetPoduct PosetPoduct@p,p2,..D computes the poduct of the posets p, p2,... in the ctego of posets nd ode peseving mps. The output is poset.

15 Cod_UGM200.nb 5 Hsse@PosetPoduct@F2, FDD ? FoestPoduct FoestPoduct@p,p2,..D computes the poduct of the foests p, p2,... in the ctego of foests nd open mps. The output is foest. Hsse@FoestPoduct@F2, FDD Acknowledgments The utho would like to thnk Silvio Icobucci who wote pt of the code (elted to poducts nd copoducts) duing his stge t the Lboto of Lnguges nd Combintoics G.-C. Rot (LIN.COM), Univesità degli Studi di Milno, unde the supevision of Pof. O. M. D'Anton. Refeences [Cod08] Pieto Cod, A theo of ptitions of ptill odeed sets, Ph.D. thesis, Univesità degli Studi di Milno, Itl (2008). [Cod09] Pieto Cod, Ptitions of finite ptill odeed set, Fom Combintoics to Philosoph: The Legc of G.-C. Rot, Spinge, New Yok (2009), [DM06] Ottvio M. D'Anton nd Vinceno M, Computing copoducts of finitel pesented Gödel lgeb, Ann. Pue Appl. Logic 42 (2006), no. -3,

Chapter 7. Kleene s Theorem. 7.1 Kleene s Theorem. The following theorem is the most important and fundamental result in the theory of FA s:

Chapter 7. Kleene s Theorem. 7.1 Kleene s Theorem. The following theorem is the most important and fundamental result in the theory of FA s: Chpte 7 Kleene s Theoem 7.1 Kleene s Theoem The following theoem is the most impotnt nd fundmentl esult in the theoy of FA s: Theoem 6 Any lnguge tht cn e defined y eithe egul expession, o finite utomt,