14 The Boole/Stone algebra of sets

Transcription

1 14 The Ble/Stne algebra f sets Lattces and Blean algebras. Gven a set A, the subsets f A admt the fllwng smple and famlar peratns n them: (ntersectn), (unn) and - (cmplementatn). If X, Y A, then X Y, X Y are als subsets f A. Wth A fxed (and suppressed n the ntatn), we wrte -X = A - X fr any X A ; f curse, -X A agan. Intersectn and unn are bnary peratns n (A), - s a unary peratn n (A) : : (A) (A) (A), : (A) (A) (A), - : (A) (A). Of curse, ntersectn and unn are defned fr any number f arguments; usng the bnary versns repeatedly, we can reprduce fnte ntersectns and unn, except the empty ntersectn and the empty unn. Fr the empty ntersectn, we take the set A tself; fr the empty unn, the empty set. What s the justfcatn? Fr any famly (A) f subsets f A, we have = {x A : fr all X, x X} and = {x A : fr sme X, x X}. Nte that the expressn fr clause " A " ; the expressn fr p. 3. s the same as that n Sectn 3, page 35 except fr the has a smlar dfference t the earler expressn n 146

2 Fr the unn, there s n actual dfference n meanng; the ld and the new expressns gve the same set. Fr the ntersectn, the same s true except fr the empty famly ; the ld expressn gves V, a nn-set; the new expressn gves A tself. Of curse, the unn f the empty famly, accrdng t the general frmula, s the empty set. It ges wthut sayng that X Y = {X, Y}, X Y = {X, Y}. The cmpste bject ( (A);,, -, A, ) (1) s an example f what we call an algebra: a set (n ths case (A) ), called the underlyng set f the algebra, wth certan partcular peratns n t (n ths case, the bnary peratns,, the unary peratn -, and the -ary peratns A, : -ary peratns are dstngushed elements f the underlyng set). Any bject f the frm (B ;,,, 1, ) wth B a set,, bth B B B, :B B, and 1, B, s an algebra smlar t (1). Speakng n very general terms, we wll seek, and at least partly fnd, prpertes f algebras f the frm (1) that dstngush them amng all the algebras smlar t them; the result wll be the ntn f Blean algebra. Fr future reference, let's say that when we dente an algebra by a sngle letter, say B, dentes the underlyng set f B. Ths, f curse, cnflcts wth the ntatn fr "cardnalty"; t s advsable t use #A fr the cardnalty f the set A when the underlyng set f an algebra s als t be used. B Let us frst lk at the basc peratns frm anther pnt f vew, namely the cntext f the pset ( (A), ). We have, fr any (A), that s the largest subset Y f A fr whch Y X fr all X : X fr all X, and f Y X fr all X, then Y ; and smlarly, 147

3 s the least subset Y f A fr whch X Y fr all X : X fr all X, and f X Y fr all X, then Y. (verfy ths statement). In general, n any pset (B, ), and fr any famly B f elements f B, a lwer bund f s any y B such that y x fr all x ; the greatest lwer bund (g.l.b), r nfmum (nf) f (f t exsts!) s the maxmum element f the set L f all lwer bunds f : y L such that y y fr all y L. (Nte that the requrement s mre than t say that y be a maxmal element f L!). The g.l.b. f s dented by ; des nt necessarly exst (n an arbtrary pset (B, ) ), but f t des, t s unquely determned by the defntn. The ntns f upper bund, least upper bund (l.u.b., supremum, sup), wth the ntatn, are defned smlarly ("dually"). [In the cntext f rdnals and well-rderngs, we have already used lub's extensvely.] Nw, ntce that what we sad abve abut ntersectns and unns amunts t ths that n the pset ( (A), ), =., exst fr all (A), and n fact =, It s wrth remarkng that the defntns f nf (sup) can be put n the fllwng frm: y y x fr all x ; y x y fr all x ; y ranges ver all the elements f the pset. Nte als that s the maxmum element f the pset (f such exsts); s the mnmum element (f exsts). We wrte 1 fr the maxmum element, fr the mnmum element (f they exst). A pset (B, ) s called a lattce f, exst fr all fnte sets B. Thus, n a lattce (B, ), there always are a maxmum element 1, a mnmum element ; mrever, fr any x, y B, x y = def {x, y}, x y = {x, y} always exst. The def 148

4 pset ( (A), ) s a lattce; n fact, t s what s called a cmplete lattce, meanng that, exst fr all B = (A). Nte the fllwng laws that always hld n any lattce: x y = y x, x y = y x (cmmutatve laws) (x y) z = x (y z), (x y) z = x (y z) (asscatve laws) x x = x, x x = x (demptent laws) x (x y) = x, x (x y) = x (absrptn laws) x 1 = x, x 1 = 1, x =, x = x. Exercses. () Verfy that the abve hld n any lattce. () Assume an algebra (B,,, 1, ) satsfyng the abve laws. Shw that there s a unque partal rderng n B that makes (B, ) a lattce n such a way that the gven,, 1, becme the lattce peratns. () Suppse that n a pset, exsts fr all sets f elements f the pset. Shw that then als always exsts. Shw that f, n ths assertn, we restrct t be a fnte set n bth ccurrences, then the resultng statement s nt always true any mre. Exercses () and () say that the cncept f lattce can be gven a purely "peratnal" ("algebrac") frmulatn. The set-theretc cmplement -X = A - X als can be gven a "lattce" descrptn. The set Y = -X s dstngushed amng all the subsets f A by the fllwng tw prpertes: Y X = A and Y X =, (verfy!). In a lattce, y s a cmplement f x f y x = 1 and y x =. In a general lattce, the cmplement f an element may nt exst, and t s als pssble that there are tw dfferent cmplements f the same element. 149

5 A partcular prperty f ( (A), ) as a lattce s that t s dstrbutve. A lattce (B, ) s dstrbutve f (x y) z = (x z) (y z) fr all x, y, z B. Indeed, the dstrbutve law s famlar fr ( (A), ) (see Assgnment 1). Exercses. (v) Shw that n a dstrbutve lattce, the dual f the dstrbutve law, that s (x y) z = (x z) (y z) hlds t. (v) Shw that n a dstrbutve lattce, every element has at mst ne cmplement. (v) Shw that any lnear rderng wth a mnmal and a maxmal element s a dstrbutve lattce. A Blean algebra s a dstrbutve lattce n whch every element has a cmplement. Of curse, ( (A), ) s a Blean algebra. One partcular Blean algebra, ( (1), ), plays a central rle n ur thery. Ths ne has tw elements: and 1 (rght?) ; nte that (1) = 2. The bnary Blean peratns are tabulated as fllws: In addtn, we have (1) =, () = 1. We call ths algebra the tw-element Blean algebra, and dente t by 2. Let us pnt ut that 2 s als cnsdered t be the algebra f truth values t=true and 15

6 f=false ; t s dentfed wth 1, f wth. Under ths dentfcatn, the abve peratns, and becme the lgcal peratns f cnjunctn ("and"), dsjunctn ("r"), and negatn ("nt"). Wth any pset B = ( B, ), we have ts ppste, B. The underlyng set f B s the same, B, as f B ; the rderng n B s the ppste f that n B : x y y x. B def It s clear that B s defned s a pset t. Mrever, t s als clear that the nf f a set n the sense f B s the same as the sup f n the sense f B, and vce versa. Thus, f B s a lattce, s s B then B. Mrever, as exercse (v) abve shws, f B s a dstrbutve lattce, s dstrbutve t. Als, the defntn f cmplement shws that the ntns f cmplement n B and B are the same. Brefly put, the ntn f "lattce", "dstrbutve lattce", and "Blean algebra" are each self-dual cncepts: f a pset falls n any f these categres, s des ts ppste Sme algebrac deas. Nte that the ntn f Blean algebra s defned n terms f the peratns,,, 1 and by denttes : the laws descrbng lattces, the dstrbutve law, and the laws defnng the cmplement. In general, an dentty, fr any knd f algebra, s an equalty f tw terms bult up f the basc peratns f the algebra, requred t hld fr all values f the varables nvlved. In the defntn f Blean algebra, we have fund sme partcular denttes that hld n the set-algebra ( (A),,,, 1, ) ; have we fund them all? As t s, ths questn s nt very ntellgent snce, e.g., 1 x = x s an dentty nt lsted abve that bvusly hlds n the set-algebra, and n fact, n all lattces, as a cnsequence f tw f the axms (why?). Hwever, we may ask: (*) s t the case that all denttes that hld n the set-algebras are cnsequences f the Blean axms, that s, are true n all Blean algebras? 151

7 Put ths way, the questn amunts t askng whether we have fund, n the Blean axms, a suffcent bass t deduce all denttes frmulated n terms,, -, 1, that are true fr sets; f the answer s "n", then there s anther, stll undscvered, essentally new dentty cncernng these set-peratns. We wll gve an affrmatve answer t the questn just asked, by deducng t frm a mre abstract therem t be stated sn. Example. The s-called De-Mrgan law: (x y) = ( x) ( y) hlds n set-algebras; n fact, t hlds, n all Blean algebras (exercse (v)). A hmmrphsm f lattces L and M, n ntatn f:l M, s a mappng f: L M between the underlyng sets that preserves the lattce peratns: f(x y) = f(x) f(y), f(x y) = f(x) f(y), f(1) = 1, f() =. These equaltes are requred t hld fr all x, y L ; f curse, n the left sdes,,, 1, refer t the lattce peratns f L, n the rght t thse f M. An embeddng f lattces s a 1-1 hmmrphsm; an smrphsm s a bjectve hmmrphsm. Exercses. (v) A lattce hmmrphsm f between Blean algebras s a Blean hmmrphsm n the sense that t als preserves cmplements: f( x) = f(x). (x) Fnd a Blean embeddng f ( (2), ) nt ( (3), ). 152

8 x y (x) Any lattce hmmrphsm preserves the partal rderng relatn: fx fy. If L, M are lattces, and f s a pset smrphsm f:( L, ) ( M, ) (.e., f s a bjectn f: L M, and x y fx fy (x, y L ) ), then f s a lattce smrphsm as well. Hwever, a pset-hmmrphsm between lattces (map preservng the rder) s nt necessarly a lattce hmmrphsm. There are the fllwng pnts t be made abut hmmrphsms and embeddngs: (1) gven a (Blean) hmmrphsm f:b C, and a Blean term t(x) bult up f varables and the symbls fr the Blean peratns, then fr any values b frm B fr the varables x we have f(t B (b)) = t C (fb) ; that s, f we frst evaluate t at b n B, then apply f, we btan the same value as when we frst apply f t each f the values n b, and then evaluate t n C at thse arguments; and (2) f an dentty s(x) = t(x) hlds n C (fr all values n C ), and f:b C s an embeddng, then the same dentty als hlds n B. (1) s a cnsequence f the defntn f "hmmrphsm"; nte that the "hmmrphsm" s defned n such a way that the assertn hld n case t s a smple term (has just ne peratn mentned n t); the general statement s prved by "nductn". (2) s a cnsequence f (1) as fllws. Suppse s(x) = t(x) hlds n C, and f:b C s an embeddng. T shw that the same dentty hlds n B, let b be arbtrary elements t evaluate the varables x. Then f(s B (b)) = s C (fb) 153

9 and f(t B (b)) = t C (fb). Snce we have s C (fb) = t C (fb) by the assumptn that the dentty hlds n C, we get B B B B that f(s (b)) = f(t (b)). Snce f s 1-1, t fllws that s (b) = t (b) as desred. Put brefly, (2) says that any dentty that hlds n an algebra hlds n any ther that can be embedded nt the gven ne. Exercse. (x) Suppse the lattce L can be embedded nt a dstrbutve lattce. Then L tself s dstrbutve. Gven a famly L I f psets, ther Cartesan prduct, L, s the pset L whse I underlyng set s L = L, and fr whch I f g f() g() fr all I. Here, f and g are arbtrary elements f L I (remember that the latter s the set f certan functns wth dman I ); n the left sde, s the rderng f L I t be defned; n the rght, refers t the rderng gven n (each) L. Exercse. (x) Verfy that L s ndeed a pset; f each L s a lattce, then s s I L ; f each L s a dstrbutve lattce, r a Blean algebra, then s s L. In fact, I I the lattce (Blean) peratns n L are defned pntwse: e.g., I (f g)() = f() g(). 154

10 (x) The prjectn mappng π : L L j j I f f(j) ne fr each j I, s a lattce hmmrphsm. (xv) Turnng t Cartesan prducts f sets, let us nte the fllwng "mappng prperty" f Cartesan prducts: fr any sets A fr I, and any further set B : the maps f:b A are n a ne-t-ne crrespndence wth famles f the I frm f :B A I. Indeed, the crrespndence, n ne drectn, asscates wth f the famly where f = π f (wth π defned as n (x) ). (xv) Nw, f the A and B are lattces (say), then the crrespndence f () gves a ne-t-ne crrespndence between hmmrphsms f:b A I and famles f hmmrphsms f the frm f :B A I. Put n anther way, t gve a hmmrphsm f:b A I s the same as t gve a famly f hmmrphsms f :B A I. When n the prduct A all the algebras A are the same, say A, we wrte A I I fr the prduct A ; A I s a pwer (the I th pwer) f A. Nte that the underlyng set f the I algebra A I, A I, s the same as I A, where n the latter the ntatn f Sectn 3, p.36 s used. The reasn why we talk abut prducts f algebras s because the pwer-set algebras ( (A), ) are all, essentally, pwers f 2, the tw-element algebra, and ths turns ut t be a useful way f lkng at pwer-set algebras. Recall the bjectn 155

11 (A) A 2 X char X. (1) A Nw, (A) and 2 are the respectve underlyng sets f the algebras ( (A), ) and A 2. We have that the mappng n (1) s an smrphsm ( (A), ) A 2. Exercse (xv): verfy ths mprtant fact. Cmbnng the last fact wth what we learned abve abut mappngs nt a prduct-algebra, we btan fr any lattce L, and any set A, the lattce hmmrphsms f:l ( (A), ) are n a ne-t-ne crrespndence wth famles f hmmrphsms f the frm f :L a 2 a A. Mrever, n ths crrespndence, f s an embeddng (1-1) f and nly f, fr every par (x, y) f dstnct elements x y f L, there s a A such that f a (x) f a (y). Exercse (xv). Verfy the last tw dsplayed assertns. Stne representatn therem fr dstrbutve lattces (and Blean algebras). Any dstrbutve lattce (hence, any Blean algebra) has an embeddng nt a pwer-set algebra. Equvalently, f L a dstrbutve lattce, and x y are arbtrary elements f L, then there s a 2-valued hmmrphsm f:l 2 such that f(x) f(y). The prf f the Stne representatn therem s the subject f the next subsectn. 156

12 Exercse (xv). Verfy that the tw versn f the therem are ndeed equvalent. Nte that the dstrbutvty cndtn n the lattce s necessary. Nte that the questn asked under (*) (at the begnnng f the present subsectn 14.2) has, as a cnsequence f the Stne representatn therem, an affrmatve answer Prme flters and ultraflters We nw set ut t prve the Stne representatn therem. Frst, we nvestgate the ntn f a 2-valued lattce hmmrphsm f:l 2. Any such f s gven by the set F = {x L : f(x)=1} ; namely, f s then the characterstc functn f X, f = char F : L 2. The questn s what prpertes F must have n rder fr char F t be a lattce hmmrphsm. We ntrduce sme standard termnlgy. Let L be a lattce. F L s a flter n L f () F 1 L F, () F F s clsed upward: x F, x y y F (x, y L ) [as a cnsequence, n () F, t wuld have been enugh t requre that F be nn-empty], and () F f x and y bth belng t F, then s des x y (x, y L ). Exercse (xx). Verfy that F L s a flter ff char F s an rder-preservng map L 2, and t als preserves and 1 [fr ths, we say that f s a meet-semlattce hmmrphsm]. A flter F n L s prme f (v) F [equvalently, F L PF L ; we say that F s a prper flter] and (v) whenever x y F, then ether x F, r y F (x, y L PF ). Exercses. (xx) The prme flters n a lattce L are n a ne-t-ne crrespndence wth the hmmrphsms L 2. (xx) Let F be a flter n the Blean algebra B. Then F s a prme flter n B ff fr any x B, exactly ne f x, x belngs t F. 157

13 In the case f a Blean algebra, we may say "ultraflter" t mean "prme flter". In vew f the refrmulatn f the ntn f 2-valued hmmrphsm as prme flter, and n vew f secnd frm f the Stne representatn therem (at the end f the secnd sectn), we nw see that the Stne representatn therem s equvalent t the fllwng statement: Fr any dstrbutve lattce L, and any par f dstnct elements x y f L, there s a prme flter P f L fr whch ne f x, y belngs t P, and the ther f x, y des nt belng t P. We are gng t shw a strnger statement, whch s als mre specfc cncernng whch f the tw gven elements can be made t belng, and whch nt t belng, t the prme flter. The strnger versn can then be used t btan ther nterestng cnsequences. The man feature f the strnger versn s a certan symmetry wth respect t "dualzng", that s, takng the ppste f the lattce n questn. Cnsder a lattce L. An deal f L s, by defntn, the same thng as a flter n L. Unravelng ths, we btan that an deal s a subset I f L such that () I, () I I L I s clsed dwnward: x I, y x y I (x, y L ), and () f bth x and y I belng t I, then s des x y (x, y L ). A prme deal f L s a prme flter f L, that s, an deal I fr whch (v) PI 1 L I, and (v) PI whenever x y I, then ether x I r y I. Prme Flter Exstence Therem (PFET). Gven any flter F and any deal I n the dstrbutve lattce L such that F and I are dsjnt: F I =, there s at least ne prme flter P n L whch cntans F as a subset and whch s dsjnt frm I : F P, I P =. Befre we turn t the prf f the PFET, let us see hw the latest frmulatn f the Stne representatn therem fllws frm t. Suppse x, y L, and x y. Then ether x y, r y x (r bth). Say, we have x y. Nw, cnsder the sets F = x = {u L : u x}, and I = y = {v L : v y}. We mmedately see def def 158

14 that x s a flter, and y s an deal (exercse). Als, they are dsjnt: f we had u x y, then x u and u y, and thus x y wuld be the case. The PFET gves a prme flter P wth x P and y P =. Then, snce x x and y y, we have that x P and y P as desred. The prf f the PFET s an applcatn f Zrn's lemma. T emphasze the character f ths prf, we slate a part f t as a separate statement. Crtern fr a prme flter. Let F be a flter, I an deal n the dstrbutve lattce L. Then any flter n L whch s maxmal amng thse flters that cntan F and dsjnt frm I s prme. Prf f the PFET frm the Crtern. Assumng the truth f the Crtern, we prceed as expected. Cnsder the set f all flters n L that cntan F as a subset and are dsjnt frm I, partally rdered by nclusn,. We apply Zrn's lemma t the pset (, ). We clam that f s any nn-empty chan n, then. Indeed, t s clear that cndtn () fr flters hlds, because s nn-empty; () s als clear. T see (), f F F F x, y, then there are F, F wth x F, y F ; snce s a chan, ether F F, r F F ; we cnclude that bth x and y belng ether t F r t F, hence, s des x y (snce F, F are flters!); but F, F are bth subsets f, thus x y belngs t as was t be shwn. As t I, f a I belnged t, then t wuld belng t an F, cntradctng F and the defntn f. The clam s verfed. The cndtn f Zrn's lemma, namely that each chan have an upper bund s almst verfed: fr each nn-empty chan, take F as an upper bund. s such an upper bund. Fr the empty chan, By Zrn's lemma, there s a maxmal element P f (, ). By the Crtern, any such maxmal element, that s, any flter maxmal amng thse flters that cntan F frm I s prme. Ths cmpletes the prf. and dsjnt Prf f the Crtern. Let P be any flter maxmal amng thse flters that cntan F and dsjnt frm I. We verfy the cndtns (v) and (v) fr P. Snce I s an PF PF 159

15 deal, I. Snce I P =, t fllws that P ; ths s (v). L L PF T see (v), assume x PF y P, and assume, cntrary t what we want, that x P and y P. We nw cnstruct a flter P[x] cntanng P {x} as a subset; we put P[x] = {u L : u s x fr sme s P}. Indeed, P[x] s a flter: cndtns () F and () F are clear; and f u, v bth belng t P[x], then there are s, t P wth u s x and v t x ; t fllws that, fr r = s t, we have u r x and v r x, and hence, u v r x (why?); ths shws that u v P[x]. Snce x P, we have P P[x]. By the maxmalty f P amng thse flters that cntan F and are dsjnt frm I, and snce clearly F P[x] (because F P ), t must be that P[x] s nt dsjnt frm I ; there s a I P[x]. The defntn f P[x] gves that there s s P such that s x a. Dng the same wth y as wth x, we get b I and t P such that t y b. Let c = a b and r = s t. Then, f curse, r x c and r y c, (why?); als, c I and r P, snce I s an deal and P s a flter. Nw [and ths s the ne pnt where we use that L s dstrbutve!], 16

16 r (x y) = (r x) (r y) c dstrbutve law means "sup" Snce we assumed that x y P, and P s a flter, t fllws that c P, n cntradctn t the fact that P I =. Ths cntradctn shws that, ndeed, x y P mples that ether x P, y P, shwng that P s a prme flter. Ths cmpletes the prf f the PFET. Exercse. (xx) A prncpal flter s ne f the frm x (fr the latter ntatn, see abve). Shw that the prncpal ultraflters f (P(A), ) are n a ne-t-ne crrespndence wth the elements f the set A. (xx) The set I rat = {q : q 1 and q s ratnal}, wth the standard rderng f the ratnal (real) numbers s a dstrbutve lattce. Gve a descrptn n famlar terms f the nn-prncpal flters f (I, ). rat (xxv) What can we say abut prme flters f a ttal rderng wth and 1 as a dstrbutve lattce? (xxv) If U and V are prme flters (ultraflters) n a Blean algebra, then U V mples U = V. * (xxv) Cnversely, f n a dstrbutve lattce L, we have that P Q mples that P = Q whenever P, Q are prme flters, then L s a Blean algebra. (xxv) Apply the PFET t shw the fllwng. Let A be any nn-empty set, and assume that s a famly f subsets f A wth the prperty that the ntersectn f any fntely many sets n s nn-empty. Shw that there s an ultraflter f ( (A), ) whch cntans. (xxv) The set A s fnte f and nly f all ultraflters f ( (A), ) are prncpal. 161