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[(c)51.1(orr)51.2(esp)51.1(onding)-339.4(to)-339.6(the)-339.5(c)51.1(ongruenc)51.1(e)]TJ /F6 1 Tf -19.7276 -1.2 TD ()Tj /F8 1 Tf 1.08 0 TD (of)Tj /F7 1 Tf 1.1755 0 TD (S)Tj /F8 1 Tf 1.0286 0 TD (is)Tj /F10 1 Tf 5.4867 -1.3501 TD ({)Tj /F6 1 Tf 0.6667 0 TD ()Tj /F10 1 Tf 1 0 TD ()Tj /F6 1 Tf 0.9444 0 TD (M\(Con)Tj /F7 1 Tf 3.25 0 TD (K)Tj /F6 1 Tf 0.9208 0 TD (\))Tj /F10 1 Tf 0.6667 0 TD (|)Tj /F6 1 Tf 0.5556 0 TD (1)Tj /F9 1 Tf 6.9738 0 0 6.9738 322.3004 526.406 Tm ()Tj /F10 1 Tf 9.9626 0 0 9.9626 331.2528 527.9004 Tm ()Tj /F6 1 Tf 1.0555 0 TD [(0)-333.3(\(\))]TJ /F10 1 Tf 2.5 0 TD (})Tj /F7 1 Tf 0.5 0 TD (,)Tj /F8 1 Tf -22.2291 -1.5602 TD [(wher)51.1(e)]TJ /F6 1 Tf 2.892 0 TD (1)Tj /F9 1 Tf 6.9738 0 0 6.9738 183.989 510.8623 Tm ()Tj /F8 1 Tf 9.9626 0 0 9.9626 194.4169 512.3567 Tm [(is)-425.9(the)-425.9(unit)-425.9(element)-425.9(of)-425.9(the)-425.9(factor)]TJ /F7 1 Tf 14.7497 0 TD (S)Tj /F9 1 Tf 6.9738 0 0 6.9738 347.472 510.8623 Tm ()Tj /F8 1 Tf 9.9626 0 0 9.9626 357.8999 512.3567 Tm (of)Tj /F7 1 Tf 1.2437 0 TD (S)Tj /F8 1 Tf 0.6709 0 TD [(.)-664.3(Henc)51.1(e,)-442.9(the)-425.9(c)51.1(ongruenc)51.1(e)]TJ -22.7629 -1.2 TD [(lattic)51.2(e)-357.8(o)0(f)]TJ /F7 1 Tf 4.1399 0 TD (S)Tj /F8 1 Tf 1.0286 0 TD [(is)-357.8(a)-357.8(nite)-357.7(Bo)51.1(ole)51.1(an)-357.8(lattic)51.2(e.)]TJ /F6 1 Tf -7.2796 -1.2 TD (\(iii\))Tj /F8 1 Tf 2.1111 0 TD [(Every)-353.7(c)51.1(ongruenc)51.2(e)]TJ /F6 1 Tf 8.0047 0 TD ()Tj /F8 1 Tf 1.076 0 TD (of)Tj /F7 1 Tf 1.1716 0 TD (K)Tj /F8 1 Tf 1.2746 0 TD [(has)-353.8(an)-353.8(extension)]TJ /F1 1 Tf 7.633 0 TD ()Tj /F6 1 Tf 0.6916 0 TD (\(\))Tj /F8 1 Tf 1.8538 0 TD [(to)-353.8(a)-353.8(c)51.1(ongruenc)51.2(e)-353.8(o)0(f)]TJ /F7 1 Tf 8.379 0 TD (S)Tj /F8 1 Tf 1.0246 0 TD [(c)51.1(orr)51.1(e-)]TJ -31.1089 -1.2 TD [(sp)51.1(onding)-357.6(to)-357.8(the)-357.8(downset)]TJ /F6 1 Tf 10.8099 0 TD ()Tj /F9 1 Tf 6.9738 0 0 6.9738 266.1928 474.9968 Tm ()Tj /F8 1 Tf 9.9626 0 0 9.9626 275.9422 476.4912 Tm (of)Tj /F6 1 Tf 1.1756 0 TD (M\(Con)Tj /F7 1 Tf 3.25 0 TD (K)Tj /F6 1 Tf 0.9208 0 TD (\))Tj /F8 1 Tf 0.3889 0 TD (.)Tj /F6 1 Tf -19.5182 -1.485 TD [(So)-289.5(w)27.8(e)-289.7(see)-289.6(no)27.8(w)-289.6(that)-289.5(a)-289.6(cubic)-289.5(extension)]TJ /F7 1 Tf 16.2444 0 TD (S)Tj /F6 1 Tf 0.9604 0 TD (of)Tj /F7 1 Tf 1.0951 0 TD (K)Tj /F6 1 Tf 1.2105 0 TD [(\(i\))-289.5(has)-289.6(a)-289.6(cubic)-289.6(congruence)-289.5(lattice)]TJ -20.7104 -1.2 TD [(\(a)-414.7(p)-27.7(o)27.7(w)27.8(e)0(r)-414.8(o)0(f)]TJ /F7 1 Tf 5.5248 0 TD (C)Tj /F9 1 Tf 6.9738 0 0 6.9738 188.8349 448.247 Tm (2)Tj /F6 1 Tf 9.9626 0 0 9.9626 193.3051 449.7415 Tm 0.4351 Tc [(,a)435.1(n)20.4()]TJ /F7 1 Tf 2.6832 0 TD 0 Tc (n)Tj /F6 1 Tf 0.6002 0 TD [(-dimensional)-414.8(cub)-27.7(e\),)-435.1(and)-414.8(\(ii\))]TJ /F7 1 Tf 13.3239 0 TD (K)Tj /F6 1 Tf 1.3356 0 TD [(and)-414.8(its)-414.8(cubic)-414.8(extension)]TJ /F7 1 Tf 10.698 0 TD (S)Tj /F6 1 Tf -35.3291 -1.2 TD [(ha)27.9(v)27.8(e)-392.8(the)-392.6(same)-392.7(n)27.8(u)0(m)27.8(b)-27.8(er)-392.6(of)-392.7(meet-irreducible)-392.6(congruences,)-407.4(and)-392.7(\(iii\))-392.7(the)-392.7(congruences)]TJ T* (of)Tj /F7 1 Tf 1.1133 0 TD (K)Tj /F6 1 Tf 1.2287 0 TD [(extend)-307.8(to)]TJ /F7 1 Tf 4.4212 0 TD (S)Tj /F6 1 Tf 0.9787 0 TD [(\(but,)-312.9(as)-307.8(a)-307.8(rule,)-313(the)-307.8(cubic)-307.8(extension)-307.9(has)-307.8(man)27.8(y)-307.8(more)-307.8(congruences\).)]TJ 2.9231 -2.469 TD (3.)Tj /F3 1 Tf 1.2778 0 TD [(Constr)22.3(ucting)-377.8(the)-377.8(poset)]TJ /F7 1 Tf 12.7116 0 TD (L)Tj /F6 1 Tf -23.4544 -1.8 TD (Let)Tj /F7 1 Tf 1.7778 0 TD (A)Tj /F6 1 Tf 1.0696 0 TD [(b)-27.7(e)-319.6(a)-319.5(nite)-319.6(lattice)-319.5(with)-319.5(zero)-319.5(0)-319.5(and)-319.6(unit)-319.5(1.)-439.9(Let)-319.5(us)-319.6(call)]TJ /F7 1 Tf 23.0466 0 TD (A)Tj /F8 1 Tf 1.0696 0 TD [(sep)51.1(ar)51.2(able)]TJ /F6 1 Tf 3.8972 0 TD [(,)-322.2(i)0(f)-319.6(i)0(t)-319.5(has)]TJ -32.0608 -1.2 TD [(an)-333.2(elemen)27.7(t)]TJ /F7 1 Tf 5.0833 0 TD (v)Tj /F6 1 Tf 0.8539 0 TD [(whic)27.8(h)-333.3(i)0(s)-333.4(a)]TJ /F8 1 Tf 4.7 0 TD [(sep)51.1(ar)51.1(ator)]TJ /F6 1 Tf 3.9867 0 TD [(,)-333.3(that)-333.2(is,)-333.4(0)]TJ /F10 1 Tf 4.8389 0 TD ()Tj /F7 1 Tf 1.0555 0 TD (v)Tj /F10 1 Tf 0.7984 0 TD ()Tj /F6 1 Tf 1.0555 0 TD (1.)Tj -21.1722 -1.2 TD [(In)-419(this)-419(section,)-440.3(for)-419.1(a)-419(nite)-419.1(p)-27.7(oset)]TJ /F7 1 Tf 15.0018 0 TD (P)Tj /F6 1 Tf 0.7809 0 TD [(,)-440.5(and)-419(a)-419.1(family)]TJ /F7 1 Tf 6.8087 0 TD (S)Tj /F14 1 Tf 6.9738 0 0 6.9738 369.8073 357.8959 Tm (p)Tj /F6 1 Tf 9.9626 0 0 9.9626 374.4119 359.3903 Tm [(,)-440.5(for)]TJ /F7 1 Tf 2.3345 0 TD (p)Tj /F10 1 Tf 0.9238 0 TD ()Tj /F7 1 Tf 1.0872 0 TD (P)Tj /F6 1 Tf 0.7809 0 TD [(,)-440.5(o)0(f)-419(separable)]TJ -29.9932 -1.2 TD [(lattices,)-333.2(w)27.7(e)-333.4(construct)-333.3(a)-333.3(p)-27.8(oset)-333.3(on)-333.3(the)-333.3(set)]TJ /F7 1 Tf 17.7472 0 TD (S)Tj /F6 1 Tf 0.9486 0 TD (=)Tj /F11 1 Tf 1.0555 0.75 TD ()Tj /F6 1 Tf 0.9445 -0.75 TD (\()Tj /F7 1 Tf 0.5555 0 TD (S)Tj /F14 1 Tf 6.9738 0 0 6.9738 344.5011 345.9407 Tm (p)Tj /F10 1 Tf 9.9626 0 0 9.9626 351.8735 347.4351 Tm (|)Tj /F7 1 Tf 0.5556 0 TD (p)Tj /F10 1 Tf 0.7809 0 TD ()Tj /F7 1 Tf 0.9444 0 TD (P)Tj /F6 1 Tf 0.9476 0 TD (\).)Tj -24.633 -1.2 TD [(W)83.4(e)-338(denote)-338.1(b)27.9(y)]TJ /F10 1 Tf 6.3473 0 TD ( )Tj /F14 1 Tf 6.9738 0 0 6.9738 208.5061 333.9856 Tm (S)Tj /F7 1 Tf 9.9626 0 0 9.9626 214.2774 335.4799 Tm (,)Tj /F10 1 Tf 0.4444 0 TD (\012)Tj /F14 1 Tf 6.9738 0 0 6.9738 225.3469 333.9856 Tm (S)Tj /F7 1 Tf 9.9626 0 0 9.9626 231.1183 335.4799 Tm (,)Tj /F10 1 Tf 0.4445 0 TD ()Tj /F14 1 Tf 6.9738 0 0 6.9738 243.2946 333.9856 Tm (S)Tj /F7 1 Tf 9.9626 0 0 9.9626 249.066 335.4799 Tm (,)Tj /F10 1 Tf 0.4445 0 TD ()Tj /F14 1 Tf 6.9738 0 0 6.9738 261.2433 333.9856 Tm (S)Tj /F6 1 Tf 9.9626 0 0 9.9626 270.382 335.4799 Tm [(the)-338(join,)-339.1(meet,)-339.2(partial)-338(ordering)-338(relation,)-339.1(and)-338.1(co)27.9(v-)]TJ -14.4248 -1.2 TD [(ering)-333.2(relation,)-333.4(resp)-27.8(ectiv)27.9(ely)83.3(,)-333.3(i)0(n)-333.4(the)-333.3(lattice)]TJ /F7 1 Tf 18.0416 0 TD (S)Tj /F6 1 Tf 0.6709 0 TD [(,)-333.3(the)-333.3(direct)-333.3(pro)-27.8(duct)-333.3(of)]TJ /F7 1 Tf 10.0611 0 TD (S)Tj /F14 1 Tf 6.9738 0 0 6.9738 419.4431 322.0304 Tm (p)Tj /F6 1 Tf 9.9626 0 0 9.9626 424.0478 323.5248 Tm [(,)-333.3(for)]TJ /F7 1 Tf 2.1417 0 TD (p)Tj /F10 1 Tf 0.7809 0 TD ()Tj /F7 1 Tf 0.9444 0 TD (P)Tj /F6 1 Tf 0.7809 0 TD (.)Tj -33.2969 -1.2 TD [(Denote)-327(b)27.8(y)-327(0)]TJ /F14 1 Tf 6.9738 0 0 6.9738 191.4989 310.0752 Tm (p)Tj /F6 1 Tf 9.9626 0 0 9.9626 196.1046 311.5696 Tm 0.3283 Tc (,1)Tj /F14 1 Tf 6.9738 0 0 6.9738 207.1243 310.0752 Tm 0 Tc (p)Tj /F6 1 Tf 9.9626 0 0 9.9626 211.729 311.5696 Tm [(,)-328.3(and)]TJ /F7 1 Tf 2.5444 0 TD (v)Tj /F14 1 Tf 6.9738 0 0 6.9738 241.9068 310.0752 Tm (p)Tj /F6 1 Tf 9.9626 0 0 9.9626 246.5116 311.5696 Tm [(,)-328.3(the)-327.1(zero,)-328.3(the)-327.1(unit,)-328.3(and)-327.1(a)-327.1(xed)-327.1(separator,)-328.2(resp)-27.8(ectiv)27.8(ely)83.3(,)]TJ -12.0288 -1.2 TD (of)Tj /F7 1 Tf 1.1388 0 TD (S)Tj /F14 1 Tf 6.9738 0 0 6.9738 144.1275 298.12 Tm (p)Tj /F6 1 Tf 9.9626 0 0 9.9626 148.7332 299.6144 Tm [(,)-333.3(for)]TJ /F7 1 Tf 2.1416 0 TD (p)Tj /F10 1 Tf 0.7809 0 TD ()Tj /F7 1 Tf 0.9445 0 TD (P)Tj /F6 1 Tf 0.7809 0 TD [(.)-444.4(I)0(f)-333.3(there)-333.3(is)-333.4(no)-333.3(am)27.8(biguit)27.8(y)83.3(,)-333.3(the)-333.3(subscripts)-333.3(will)-333.3(b)-27.8(e)-333.3(dropp)-27.8(ed.)]TJ -5.6622 -1.2 TD [(W)83.4(e)-419.6(need)-419.7(some)-419.6(notation)-419.6(in)]TJ /F7 1 Tf 12.1591 0 TD (S)Tj /F6 1 Tf 0.6708 0 TD [(.)-703.3(The)-419.6(elemen)27.8(ts)-419.7(of)]TJ /F7 1 Tf 8.5232 0 TD (S)Tj /F6 1 Tf 1.0905 0 TD [(will)-419.5(b)-27.8(e)-419.6(b)-27.8(old)-419.6(face)-419.5(lo)27.8(w)27.7(e)0(r)-419.6(case)]TJ -23.6436 -1.2 TD [(letters.)-512.1(An)-355.8(elemen)27.8(t)]TJ /F1 1 Tf 8.8989 0 TD (s)Tj /F10 1 Tf 0.769 0 TD ()Tj /F7 1 Tf 0.9821 0 TD (S)Tj /F6 1 Tf 1.0267 0 TD [(is)-356(written)-355.8(in)-356(the)-355.9(form)]TJ /F10 1 Tf 9.874 0 TD ()Tj /F7 1 Tf 0.3889 0 TD (s)Tj /F14 1 Tf 6.9738 0 0 6.9738 349.9198 274.2097 Tm (p)Tj /F10 1 Tf 9.9626 0 0 9.9626 354.5245 275.7041 Tm ()Tj /F14 1 Tf 6.9738 0 0 6.9738 358.399 274.2097 Tm (p)Tj /F16 1 Tf 0.5889 0 TD ()Tj /F14 1 Tf 0.7699 0 TD (P)Tj /F6 1 Tf 9.9626 0 0 9.9626 374.5175 275.7041 Tm [(.)-512.1(W)83.3(e)-355.9(write)]TJ /F1 1 Tf 5.1156 0 TD (s)Tj /F14 1 Tf 6.9738 0 0 6.9738 430.0015 274.2097 Tm (p)Tj /F6 1 Tf 9.9626 0 0 9.9626 438.1529 275.7041 Tm (for)Tj /F7 1 Tf 1.5531 0 TD (s)Tj /F14 1 Tf 6.9738 0 0 6.9738 458.2964 274.2097 Tm (p)Tj /F6 1 Tf 9.9626 0 0 9.9626 462.9011 275.7041 Tm 0.5121 Tc [(.F)595.4(o)512.1(r)]TJ /F7 1 Tf -33.7489 -1.2 TD 0 Tc (q)Tj /F10 1 Tf 0.76 0 TD ()Tj /F7 1 Tf 0.9445 0 TD (P)Tj /F6 1 Tf 0.7809 0 TD [(,)-328.6(let)]TJ /F1 1 Tf 2.0451 0 TD (u)Tj /F14 1 Tf 6.9738 0 0 6.9738 178.1738 267.3644 Tm (q)Tj /F10 1 Tf 9.9626 0 0 9.9626 185.341 263.749 Tm ()Tj /F7 1 Tf 0.9444 0 TD (S)Tj /F6 1 Tf 0.9984 0 TD [(b)-27.7(e)-327.6(dened)-327.6(b)27.8(y)-327.5(\()]TJ /F1 1 Tf 6.566 0 TD (u)Tj /F14 1 Tf 6.9738 0 0 6.9738 276.4762 267.3644 Tm (q)Tj /F6 1 Tf 9.9626 0 0 9.9626 280.8757 263.749 Tm (\))Tj /F14 1 Tf 6.9738 0 0 6.9738 284.7502 262.2545 Tm (q)Tj /F6 1 Tf 9.9626 0 0 9.9626 291.9173 263.749 Tm [(=)-277.8(1)-327.5(and)-327.6(otherwise,)-328.6(\()]TJ /F1 1 Tf 8.9366 0 TD (u)Tj /F14 1 Tf 6.9738 0 0 6.9738 387.3146 267.3644 Tm (q)Tj /F6 1 Tf 9.9626 0 0 9.9626 391.7141 263.749 Tm (\))Tj /F14 1 Tf 6.9738 0 0 6.9738 395.5886 262.2545 Tm (p)Tj /F6 1 Tf 9.9626 0 0 9.9626 402.9609 263.749 Tm 0.2778 Tc [(=0)277.8(.)-164.7(F)361.2(o)277.8(r)]TJ /F7 1 Tf 4.0645 0 TD 0 Tc (q)Tj /F10 1 Tf 0.7601 0 TD ()Tj /F7 1 Tf 0.9444 0 TD (P)Tj /F6 1 Tf 0.7809 0 TD [(,)-328.7(let)]TJ /F1 1 Tf -34.2823 -1.2 TD (v)Tj /F14 1 Tf 6.9738 0 0 6.9738 132.8787 255.4092 Tm (q)Tj /F10 1 Tf 9.9626 0 0 9.9626 140.0458 251.7937 Tm ()Tj /F7 1 Tf 0.9444 0 TD (S)Tj /F6 1 Tf 1.0042 0 TD [(b)-27.7(e)-333.4(dened)-333.3(b)27.8(y)-333.3(\()]TJ /F1 1 Tf 6.5833 0 TD (v)Tj /F14 1 Tf 6.9738 0 0 6.9738 231.2518 255.4092 Tm (q)Tj /F6 1 Tf 9.9626 0 0 9.9626 235.6523 251.7937 Tm (\))Tj /F14 1 Tf 6.9738 0 0 6.9738 239.5268 250.2993 Tm (q)Tj /F6 1 Tf 9.9626 0 0 9.9626 246.6939 251.7937 Tm (=)Tj /F7 1 Tf 1.0555 0 TD (v)Tj /F6 1 Tf 0.8539 0 TD [(and)-333.4(otherwise,)-333.3(\()]TJ /F1 1 Tf 7.0639 0 TD (v)Tj /F14 1 Tf 6.9738 0 0 6.9738 342.2984 255.4092 Tm (q)Tj /F6 1 Tf 9.9626 0 0 9.9626 346.6979 251.7937 Tm (\))Tj /F14 1 Tf 6.9738 0 0 6.9738 350.5723 250.2993 Tm (p)Tj /F6 1 Tf 9.9626 0 0 9.9626 357.9447 251.7937 Tm 0.2777 Tc [(=0)277.7(.)]TJ -22.0139 -1.2 TD 0 Tc (Let)Tj /F7 1 Tf 1.774 0 TD (B)Tj /F6 1 Tf 1.1245 0 TD [(b)-27.7(e)-315.8(the)-315.7(sublattice)-315.8(of)]TJ /F7 1 Tf 8.713 0 TD (S)Tj /F6 1 Tf 0.9866 0 TD [(generated)-315.7(b)27.8(y)]TJ /F10 1 Tf 5.912 0 TD ({)Tj /F1 1 Tf 0.6667 0 TD (u)Tj /F14 1 Tf 6.9738 0 0 6.9738 336.0448 243.454 Tm (p)Tj /F10 1 Tf 9.9626 0 0 9.9626 343.4172 239.8386 Tm (|)Tj /F7 1 Tf 0.5555 0 TD (p)Tj /F10 1 Tf 0.7809 0 TD ()Tj /F7 1 Tf 0.9445 0 TD (P)Tj /F10 1 Tf 0.9475 0 TD (})Tj /F6 1 Tf 0.5 0 TD [(.)-438.6(W)83.4(e)-315.8(call)]TJ /F7 1 Tf 4.2368 0 TD (B)Tj /F6 1 Tf 1.1244 0 TD (the)Tj /F8 1 Tf 1.7047 0 TD (skeleton)Tj /F6 1 Tf -32.55 -1.2 TD (of)Tj /F7 1 Tf 1.2924 0 TD (S)Tj /F6 1 Tf 0.6708 0 TD [(;)-563.7(i)0(t)-486.9(i)0(s)-486.9(a)-486.9(b)-27.8(o)-27.7(olean)-486.9(sublattice)-486.9(of)]TJ /F7 1 Tf 14.0241 0 TD (S)Tj /F6 1 Tf 1.1577 0 TD (with)Tj /F7 1 Tf 2.4314 0 TD (n)Tj /F6 1 Tf 1.0871 0 TD [(atoms.)-905.1(F)83.4(o)0(r)-486.9(a)-486.9(subset)]TJ /F7 1 Tf 9.9548 0 TD (Q)Tj /F6 1 Tf 1.2774 0 TD (of)Tj /F7 1 Tf 1.2925 0 TD (P)Tj /F6 1 Tf 0.7809 0 TD [(,)-525.3(set)]TJ /F1 1 Tf -33.9691 -1.2 TD (u)Tj /F14 1 Tf 6.9738 0 0 6.9738 133.0371 219.5437 Tm (Q)Tj /F6 1 Tf 9.9626 0 0 9.9626 144.0827 215.9282 Tm (=)Tj /F11 1 Tf 1.2068 0.75 TD ()Tj /F14 1 Tf 6.9738 0 0 6.9738 164.4075 212.9395 Tm (S)Tj /F10 1 Tf 9.9626 0 0 9.9626 171.8396 215.9282 Tm ({)Tj /F1 1 Tf 0.5 0 TD (u)Tj /F14 1 Tf 6.9738 0 0 6.9738 183.186 219.5437 Tm (p)Tj /F10 1 Tf 9.9626 0 0 9.9626 192.0638 215.9282 Tm (|)Tj /F7 1 Tf 0.7067 0 TD (p)Tj /F10 1 Tf 0.9321 0 TD ()Tj /F7 1 Tf 1.0956 0 TD (Q)Tj /F10 1 Tf 0.7905 0 TD (})Tj /F6 1 Tf 0.9241 0 TD [(with)-424(complemen)27.8(t)-424(\()]TJ /F1 1 Tf 8.4314 0 TD (u)Tj /F14 1 Tf 6.9738 0 0 6.9738 326.7517 219.5437 Tm (Q)Tj /F6 1 Tf 9.9626 0 0 9.9626 333.5233 215.9282 Tm (\))Tj /F16 1 Tf 6.9738 0 0 6.9738 337.3978 219.5437 Tm ()Tj /F6 1 Tf 9.9626 0 0 9.9626 344.4662 215.9282 Tm (=)Tj /F1 1 Tf 1.2067 0 TD (u)Tj /F14 1 Tf 6.9738 0 0 6.9738 362.8533 219.5437 Tm (P)Tj /F16 1 Tf 0.8811 0 TD ()Tj /F14 1 Tf 0.8929 0 TD (Q)Tj /F6 1 Tf 9.9626 0 0 9.9626 386.2216 215.9282 Tm (in)Tj /F7 1 Tf 1.2574 0 TD (B)Tj /F6 1 Tf 1.2327 0 TD [(\(and)-424(in)]TJ /F7 1 Tf 3.6814 0 TD (S)Tj /F6 1 Tf 0.6708 0 TD [(\).)-716.5(The)]TJ -32.8945 -1.2 TD [(elemen)27.8(ts)-333.3(of)]TJ /F7 1 Tf 5.2277 0 TD (B)Tj /F6 1 Tf 1.142 0 TD [(are)-333.4(blac)27.8(klled)-333.3(in)-333.4(Figures)-333.3(1)-333.3(and)-333.4(2.)]TJ -5.1697 -1.2 TD [(No)27.8(w)-333.3(w)27.7(e)-333.4(come)-333.3(to)-333.3(the)-333.4(crucial)-333.3(denition)-333.3(of)-333.3(this)-333.3(pap)-27.8(er:)]TJ T* [(On)-333.3(the)-333.3(set)]TJ /F7 1 Tf 4.95 0 TD (S)Tj /F6 1 Tf 0.6708 0 TD [(,)-333.3(w)27.8(e)-333.4(dene)-333.3(a)-333.3(binary)-333.3(relation)]TJ /F10 1 Tf 12.6167 0 TD ()Tj /F6 1 Tf 0.7778 0 TD [(.)-444.4(Let)]TJ /F1 1 Tf 2.5138 0 TD (a)Tj /F6 1 Tf 0.8368 0 TD (=)Tj /F10 1 Tf 1.0556 0 TD ()Tj /F7 1 Tf 0.3889 0 TD (a)Tj /F14 1 Tf 6.9738 0 0 6.9738 381.1089 178.5684 Tm (p)Tj /F10 1 Tf 9.9626 0 0 9.9626 385.7136 180.0627 Tm ()Tj /F14 1 Tf 6.9738 0 0 6.9738 389.588 178.5684 Tm (p)Tj /F16 1 Tf 0.5889 0 TD ()Tj /F14 1 Tf 0.7699 0 TD (P)Tj /F7 1 Tf 9.9626 0 0 9.9626 405.7066 180.0627 Tm (,)Tj /F1 1 Tf 0.4444 0 TD 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(p)Tj
/F16 1 Tf
6.9738 0 0 6.9738 313.0202 640.8458 Tm
()Tj
/F8 1 Tf
9.9626 0 0 9.9626 319.3793 637.2304 Tm
[(is)-357.8(an)]TJ
/F10 1 Tf
2.5045 0 TD
({)Tj
/F1 1 Tf
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(a)Tj
/F7 1 Tf
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(,)Tj
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(b)Tj
/F10 1 Tf
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(})Tj
/F8 1 Tf
0.5 0 TD
[(-fork,)-357.8(or)]TJ
/F6 1 Tf
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(\(2\))Tj
/F7 1 Tf
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(b)Tj
/F14 1 Tf
6.9738 0 0 6.9738 312.2829 621.3898 Tm
(p)Tj
/F10 1 Tf
9.9626 0 0 9.9626 319.6553 622.8842 Tm
(\015)Tj
/F7 1 Tf
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(a)Tj
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6.9738 0 0 6.9738 335.4381 621.3898 Tm
(p)Tj
/F8 1 Tf
9.9626 0 0 9.9626 343.6075 622.8842 Tm
(and)Tj
/F7 1 Tf
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(b)Tj
/F14 1 Tf
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(p)Tj
/F15 1 Tf
4.9813 0 0 4.9813 371.3395 623.3823 Tm
()Tj
/F13 1 Tf
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()Tj
/F7 1 Tf
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(a)Tj
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6.9738 0 0 6.9738 393.082 621.3898 Tm
(p)Tj
/F15 1 Tf
4.9813 0 0 4.9813 397.1885 623.3823 Tm
()Tj
/F8 1 Tf
9.9626 0 0 9.9626 400.3816 622.8842 Tm
0.3578 Tc
[(,o)357.8(r)]TJ
/F6 1 Tf
-11.078 -1.44 TD
0 Tc
(\(3\))Tj
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1.806 0 TD
(a)Tj
/F14 1 Tf
6.9738 0 0 6.9738 313.2742 607.0436 Tm
(p)Tj
/F10 1 Tf
9.9626 0 0 9.9626 320.6465 608.538 Tm
(\015)Tj
/F7 1 Tf
1.0555 0 TD
(b)Tj
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(p)Tj
/F8 1 Tf
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(and)Tj
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(a)Tj
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6.9738 0 0 6.9738 368.2232 607.0436 Tm
(p)Tj
/F15 1 Tf
4.9813 0 0 4.9813 372.3297 609.0361 Tm
()Tj
/F13 1 Tf
9.9626 0 0 9.9626 378.2904 608.538 Tm
()Tj
/F7 1 Tf
1.0555 0 TD
(b)Tj
/F14 1 Tf
6.9738 0 0 6.9738 393.082 607.0436 Tm
(p)Tj
/F15 1 Tf
4.9813 0 0 4.9813 397.1885 609.0361 Tm
()Tj
/F8 1 Tf
9.9626 0 0 9.9626 400.3816 608.538 Tm
(;)Tj
/F7 1 Tf
-16.3935 -1.44 TD
(a)Tj
/F14 1 Tf
6.9738 0 0 6.9738 242.3253 592.6974 Tm
(p)Tj
/F10 1 Tf
9.9626 0 0 9.9626 249.1437 594.1918 Tm
(\012)Tj
/F7 1 Tf
0.8889 0 TD
(b)Tj
/F14 1 Tf
6.9738 0 0 6.9738 262.2755 592.6974 Tm
(p)Tj
/F7 1 Tf
9.9626 0 0 9.9626 266.8802 594.1918 Tm
(,)Tj
/F8 1 Tf
1.3 0 TD
(otherwise)Tj
/F7 1 Tf
4.0761 0 TD
(.)Tj
/F8 1 Tf
-19.4494 -2.31 TD
[(Pr)51.2(o)51.1(of.)]TJ
/F6 1 Tf
3.1333 0 TD
[(Recall)-272.7(that)]TJ
/F7 1 Tf
5.0594 0 TD
(v)Tj
/F6 1 Tf
0.7934 0 TD
[(is)-272.8(separable,)-284.9(and)-272.8(therefore)-272.8(join-irreducible,)-284.7(hence)]TJ
/F7 1 Tf
21.4638 0 TD
(a)Tj
/F14 1 Tf
6.9738 0 0 6.9738 435.3006 569.6837 Tm
(p)Tj
/F10 1 Tf
9.9626 0 0 9.9626 440.9126 571.1781 Tm
( )Tj
/F7 1 Tf
0.7678 0 TD
(b)Tj
/F14 1 Tf
6.9738 0 0 6.9738 452.8379 569.6837 Tm
(p)Tj
/F6 1 Tf
9.9626 0 0 9.9626 460.2102 571.1781 Tm
(=)Tj
/F7 1 Tf
1.0555 0 TD
(v)Tj
/F6 1 Tf
0.7934 0 TD
(is)Tj
-35.3277 -1.2 TD
[(equiv)55.7(alen)27.8(t)-323.8(t)0(o)]TJ
/F10 1 Tf
5.953 0 TD
()Tj
/F7 1 Tf
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(a)Tj
/F14 1 Tf
6.9738 0 0 6.9738 195.1213 557.7285 Tm
(p)Tj
/F7 1 Tf
9.9626 0 0 9.9626 199.726 559.2229 Tm
0.1667 Tc
(,b)Tj
/F14 1 Tf
6.9738 0 0 6.9738 208.4294 557.7285 Tm
0 Tc
(p)Tj
/F10 1 Tf
9.9626 0 0 9.9626 213.0351 559.2229 Tm
0.2778 Tc
[({)277.8()]TJ
/F6 1 Tf
2.5 0 TD
0 Tc
(0)Tj
/F7 1 Tf
0.5 0 TD
0.1666 Tc
(,v)Tj
/F10 1 Tf
0.965 0 TD
0 Tc
()Tj
/F7 1 Tf
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(,)Tj
/F10 1 Tf
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()Tj
/F7 1 Tf
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(v,)Tj
/F6 1 Tf
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0 Tc
(0)Tj
/F10 1 Tf
0.5 0 TD
()Tj
/F7 1 Tf
0.3889 0 TD
(,)Tj
/F10 1 Tf
0.4444 0 TD
()Tj
/F7 1 Tf
0.3889 0 TD
0.0359 Tc
[(v,)-130.7(v)]TJ
/F10 1 Tf
1.4856 0 TD
0 Tc
(})Tj
/F6 1 Tf
1.2127 0 TD
[(and)-323.7(similarly)83.4(,)]TJ
/F7 1 Tf
6.2133 0 TD
(a)Tj
/F14 1 Tf
6.9738 0 0 6.9738 385.5352 557.7285 Tm
(p)Tj
/F10 1 Tf
9.9626 0 0 9.9626 392.1624 559.2229 Tm
(\012)Tj
/F7 1 Tf
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(b)Tj
/F14 1 Tf
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(p)Tj
/F6 1 Tf
9.9626 0 0 9.9626 412.4752 559.2229 Tm
(=)Tj
/F7 1 Tf
1.0556 0 TD
(v)Tj
/F6 1 Tf
0.8443 0 TD
[(is)-323.8(equiv)55.6(alen)27.8(t)]TJ
-30.5873 -1.2 TD
(to)Tj
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()Tj
/F7 1 Tf
0.3889 0 TD
(a)Tj
/F14 1 Tf
6.9738 0 0 6.9738 148.8378 545.7734 Tm
(p)Tj
/F7 1 Tf
9.9626 0 0 9.9626 153.4426 547.2678 Tm
0.1667 Tc
(,b)Tj
/F14 1 Tf
6.9738 0 0 6.9738 162.1459 545.7734 Tm
0 Tc
(p)Tj
/F10 1 Tf
9.9626 0 0 9.9626 166.7517 547.2678 Tm
0.4197 Tc
[({)419.7()]TJ
/F6 1 Tf
2.7838 0 TD
0 Tc
(1)Tj
/F7 1 Tf
0.5 0 TD
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(,v)Tj
/F10 1 Tf
0.965 0 TD
0 Tc
()Tj
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(,)Tj
/F10 1 Tf
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()Tj
/F7 1 Tf
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(v,)Tj
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(1)Tj
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()Tj
/F7 1 Tf
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(,)Tj
/F10 1 Tf
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()Tj
/F7 1 Tf
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0.0359 Tc
[(v,)-130.8(v)]TJ
/F10 1 Tf
1.4857 0 TD
0 Tc
(})Tj
/F6 1 Tf
0.8889 0 TD
[(.)-699.8(Also)-418.5(notice)-418.5(that)-418.6(if)]TJ
/F7 1 Tf
9.6016 0 TD
(a)Tj
/F14 1 Tf
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(p)Tj
/F6 1 Tf
9.9626 0 0 9.9626 381.3968 547.2678 Tm
(=)Tj
/F7 1 Tf
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(b)Tj
/F14 1 Tf
6.9738 0 0 6.9738 397.602 545.7734 Tm
(p)Tj
/F6 1 Tf
9.9626 0 0 9.9626 406.3881 547.2678 Tm
(=)Tj
/F7 1 Tf
1.1975 0 TD
(v)Tj
/F6 1 Tf
0.9391 0 TD
[(and)-418.5(Case)-418.5(\(1\))]TJ
-30.213 -1.2 TD
[(o)-27.7(ccurs,)-361.8(then)]TJ
/F7 1 Tf
5.6985 0 TD
(p)Tj
/F6 1 Tf
0.8593 0 TD
[(is)-356.1(an)]TJ
/F10 1 Tf
2.44 0 TD
({)Tj
/F1 1 Tf
0.5 0 TD
(a)Tj
/F7 1 Tf
0.559 0 TD
(,)Tj
/F1 1 Tf
0.4445 0 TD
(b)Tj
/F10 1 Tf
0.6389 0 TD
(})Tj
/F6 1 Tf
0.5 0 TD
[(-fork;)-367.5(in)-356.2(Cases)-356.1(\(2\))-356.1(and)-356.2(\(3\),)-361.8(w)27.8(e)-356.2(m)27.7(ust)-356.1(ha)27.8(v)27.8(e)]TJ
/F7 1 Tf
18.5474 0 TD
(p)Tj
/F16 1 Tf
6.9738 0 0 6.9738 432.4334 538.928 Tm
()Tj
/F7 1 Tf
9.9626 0 0 9.9626 438.3741 535.3126 Tm
0.3158 Tc
(>p)Tj
/F6 1 Tf
1.5967 0 TD
0 Tc
[(.)-512.9(Let)]TJ
/F1 1 Tf
2.6051 0 TD
(c)Tj
/F6 1 Tf
-35.4888 -1.2 TD
[(b)-27.7(e)-334.8(the)-334.7(elemen)27.8(t)-334.7(dened)-334.8(in)-334.8(the)-334.7(join)-334.7(form)27.8(ula.)-448.6(T)83.3(o)-334.8(see)-334.7(that)]TJ
/F1 1 Tf
24.4154 0 TD
(a)Tj
/F10 1 Tf
0.8391 0 TD
()Tj
/F1 1 Tf
1.0579 0 TD
(c)Tj
/F6 1 Tf
0.8458 0 TD
[(\(using)-334.8(the)-334.7(denition)]TJ
-27.1582 -1.2 TD
[(\(P\))-349.8(of)]TJ
/F10 1 Tf
2.9634 0 TD
()Tj
/F6 1 Tf
0.7778 0 TD
[(\),)-353.9(rst)-349.8(note)-349.8(that)]TJ
/F1 1 Tf
7.5228 0 TD
(a)Tj
/F10 1 Tf
0.8643 0 TD
()Tj
/F14 1 Tf
6.9738 0 0 6.9738 255.2518 509.9078 Tm
(S)Tj
/F1 1 Tf
9.9626 0 0 9.9626 264.0638 511.4023 Tm
(a)Tj
/F10 1 Tf
0.7922 0 TD
( )Tj
/F14 1 Tf
6.9738 0 0 6.9738 278.5983 509.9078 Tm
(S)Tj
/F1 1 Tf
9.9626 0 0 9.9626 286.6929 511.4023 Tm
(b)Tj
/F10 1 Tf
0.9441 0 TD
()Tj
/F14 1 Tf
6.9738 0 0 6.9738 303.8476 509.9078 Tm
(S)Tj
/F1 1 Tf
9.9626 0 0 9.9626 312.6595 511.4023 Tm
(c)Tj
/F6 1 Tf
0.5111 0 TD
[(,)-354(s)0(o)-349.8(that)]TJ
/F1 1 Tf
4.0592 0 TD
(a)Tj
/F10 1 Tf
0.8642 0 TD
()Tj
/F14 1 Tf
6.9738 0 0 6.9738 374.5504 509.9078 Tm
(S)Tj
/F1 1 Tf
9.9626 0 0 9.9626 383.3624 511.4023 Tm
(c)Tj
/F6 1 Tf
0.5111 0 TD
[(.)-493.9(Let)]TJ
/F7 1 Tf
2.5798 0 TD
(a)Tj
/F14 1 Tf
6.9738 0 0 6.9738 419.4222 509.9078 Tm
(p)Tj
/F6 1 Tf
9.9626 0 0 9.9626 427.0685 511.4023 Tm
(=)Tj
/F7 1 Tf
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(c)Tj
/F14 1 Tf
6.9738 0 0 6.9738 442.1689 509.9078 Tm
(p)Tj
/F6 1 Tf
9.9626 0 0 9.9626 449.8152 511.4023 Tm
(=)Tj
/F7 1 Tf
1.083 0 TD
(v)Tj
/F6 1 Tf
0.8704 0 TD
(and)Tj
/F7 1 Tf
-34.3888 -1.2 TD
(p)Tj
/F16 1 Tf
6.9738 0 0 6.9738 131.6852 503.0625 Tm
()Tj
/F7 1 Tf
9.9626 0 0 9.9626 137.2473 499.4471 Tm
0.2778 Tc
(>p)Tj
/F6 1 Tf
1.5587 0 TD
0 Tc
[(;)-333.3(w)27.8(e)-333.4(m)27.7(ust)-333.3(sho)27.8(w)-333.4(that)]TJ
/F7 1 Tf
9.2055 0 TD
(a)Tj
/F14 1 Tf
6.9738 0 0 6.9738 249.7534 497.9527 Tm
(p)Tj
/F15 1 Tf
4.9813 0 0 4.9813 253.861 499.9452 Tm
()Tj
/F6 1 Tf
9.9626 0 0 9.9626 259.8206 499.4471 Tm
(=)Tj
/F7 1 Tf
1.0556 0 TD
(c)Tj
/F14 1 Tf
6.9738 0 0 6.9738 274.6481 497.9527 Tm
(p)Tj
/F15 1 Tf
4.9813 0 0 4.9813 278.7557 499.9452 Tm
()Tj
/F6 1 Tf
9.9626 0 0 9.9626 281.9487 499.4471 Tm
(.)Tj
-14.3858 -1.2 TD
[(No)27.8(w)]TJ
/F7 1 Tf
2.3725 0 TD
(a)Tj
/F14 1 Tf
6.9738 0 0 6.9738 167.5307 485.9975 Tm
(p)Tj
/F6 1 Tf
9.9626 0 0 9.9626 176.4762 487.4919 Tm
(=)Tj
/F7 1 Tf
1.2135 0 TD
(c)Tj
/F14 1 Tf
6.9738 0 0 6.9738 192.8777 485.9975 Tm
(p)Tj
/F6 1 Tf
9.9626 0 0 9.9626 201.8232 487.4919 Tm
(=)Tj
/F7 1 Tf
1.2135 0 TD
(v)Tj
/F6 1 Tf
0.9487 0 TD
[(implies)-428.1(that)]TJ
/F7 1 Tf
5.7506 0 TD
(a)Tj
/F14 1 Tf
6.9738 0 0 6.9738 285.9218 485.9975 Tm
(p)Tj
/F6 1 Tf
9.9626 0 0 9.9626 294.8672 487.4919 Tm
(=)Tj
/F7 1 Tf
1.2135 0 TD
(a)Tj
/F14 1 Tf
6.9738 0 0 6.9738 312.2231 485.9975 Tm
(p)Tj
/F10 1 Tf
9.9626 0 0 9.9626 319.6712 487.4919 Tm
( )Tj
/F7 1 Tf
0.9521 0 TD
(b)Tj
/F14 1 Tf
6.9738 0 0 6.9738 333.4326 485.9975 Tm
(p)Tj
/F6 1 Tf
9.9626 0 0 9.9626 342.3781 487.4919 Tm
(=)Tj
/F7 1 Tf
1.2135 0 TD
(c)Tj
/F14 1 Tf
6.9738 0 0 6.9738 358.7786 485.9975 Tm
(p)Tj
/F6 1 Tf
9.9626 0 0 9.9626 367.725 487.4919 Tm
(=)Tj
/F7 1 Tf
1.2135 0 TD
(v)Tj
/F6 1 Tf
0.5206 0 TD
[(,)-451.7(and)-428.1(so)]TJ
/F7 1 Tf
4.0913 0 TD
(b)Tj
/F14 1 Tf
6.9738 0 0 6.9738 430.0364 485.9975 Tm
(p)Tj
/F10 1 Tf
9.9626 0 0 9.9626 438.9828 487.4919 Tm
()Tj
/F7 1 Tf
1.2135 0 TD
(a)Tj
/F14 1 Tf
6.9738 0 0 6.9738 456.3377 485.9975 Tm
(p)Tj
/F6 1 Tf
9.9626 0 0 9.9626 465.2842 487.4919 Tm
(=)Tj
/F7 1 Tf
1.2135 0 TD
(v)Tj
/F6 1 Tf
0.5206 0 TD
(.)Tj
-35.7222 -1.2 TD
[(Hence,)-324.1(for)-321.9(all)]TJ
/F7 1 Tf
6.1373 0 TD
(q)Tj
/F10 1 Tf
0.7601 0 TD
(\015)Tj
/F7 1 Tf
1.0555 0 TD
(p)Tj
/F6 1 Tf
0.5031 0 TD
[(,)-324.2(none)-321.8(of)-321.8(Cases)-321.8(\(1\)\(3\))-321.9(o)-27.7(ccur.)-440.7(That)-321.8(is,)-324.2(for)-321.8(all)]TJ
/F7 1 Tf
20.0032 0 TD
(q)Tj
/F10 1 Tf
0.76 0 TD
(\015)Tj
/F7 1 Tf
1.0556 0 TD
(p)Tj
/F6 1 Tf
0.5031 0 TD
(,)Tj
/F7 1 Tf
0.6019 0 TD
(q)Tj
/F6 1 Tf
0.8042 0 TD
[(is)-321.9(not)-321.8(an)]TJ
/F10 1 Tf
-32.184 -1.2 TD
({)Tj
/F1 1 Tf
0.5 0 TD
(a)Tj
/F7 1 Tf
0.559 0 TD
(,)Tj
/F1 1 Tf
0.4444 0 TD
(b)Tj
/F10 1 Tf
0.6389 0 TD
(})Tj
/F6 1 Tf
0.5 0 TD
[(-fork)-323.8(\(Case)-323.8(\(1\)\))-323.8(and)]TJ
/F7 1 Tf
9.0815 0 TD
(b)Tj
/F14 1 Tf
6.9738 0 0 6.9738 247.7489 462.0872 Tm
(q)Tj
/F10 1 Tf
9.9626 0 0 9.9626 254.9161 463.5816 Tm
()Tj
/F7 1 Tf
1.0555 0 TD
(a)Tj
/F14 1 Tf
6.9738 0 0 6.9738 270.6979 462.0872 Tm
(q)Tj
/F6 1 Tf
9.9626 0 0 9.9626 278.3243 463.5816 Tm
[(\(Case)-323.8(\(2\)\).)-441.3(In)-323.8(Case)-323.8(\(3\),)-325.7(using)-323.8(that)]TJ
/F7 1 Tf
15.4307 0 TD
(b)Tj
/F14 1 Tf
6.9738 0 0 6.9738 436.3307 462.0872 Tm
(p)Tj
/F10 1 Tf
9.9626 0 0 9.9626 443.7031 463.5816 Tm
()Tj
/F7 1 Tf
1.0555 0 TD
(a)Tj
/F14 1 Tf
6.9738 0 0 6.9738 459.4849 462.0872 Tm
(p)Tj
/F6 1 Tf
9.9626 0 0 9.9626 466.8573 463.5816 Tm
(=)Tj
/F7 1 Tf
1.0556 0 TD
(v)Tj
/F6 1 Tf
0.5206 0 TD
(,)Tj
-35.7222 -1.2 TD
(if)Tj
/F7 1 Tf
0.9166 0 TD
(a)Tj
/F14 1 Tf
6.9738 0 0 6.9738 141.071 450.132 Tm
(p)Tj
/F10 1 Tf
9.9626 0 0 9.9626 148.4433 451.6264 Tm
()Tj
/F7 1 Tf
1.0556 0 TD
(b)Tj
/F14 1 Tf
6.9738 0 0 6.9738 163.2349 450.132 Tm
(p)Tj
/F6 1 Tf
9.9626 0 0 9.9626 167.8406 451.6264 Tm
[(,)-333.3(then)]TJ
/F7 1 Tf
2.8889 0 TD
(a)Tj
/F14 1 Tf
6.9738 0 0 6.9738 201.8869 450.132 Tm
(p)Tj
/F6 1 Tf
9.9626 0 0 9.9626 209.2603 451.6264 Tm
(=)Tj
/F7 1 Tf
1.0555 0 TD
(b)Tj
/F14 1 Tf
6.9738 0 0 6.9738 224.0518 450.132 Tm
(p)Tj
/F6 1 Tf
9.9626 0 0 9.9626 231.4241 451.6264 Tm
(=)Tj
/F7 1 Tf
1.0555 0 TD
(v)Tj
/F6 1 Tf
0.5206 0 TD
[(,)-333.3(and)-333.4(since)]TJ
/F7 1 Tf
5.0056 0 TD
(p)Tj
/F6 1 Tf
0.8364 0 TD
[(is)-333.4(not)-333.3(an)]TJ
/F10 1 Tf
4.1723 0 TD
({)Tj
/F1 1 Tf
0.5 0 TD
(a)Tj
/F7 1 Tf
0.559 0 TD
(,)Tj
/F1 1 Tf
0.4444 0 TD
(b)Tj
/F10 1 Tf
0.6389 0 TD
(})Tj
/F6 1 Tf
0.5 0 TD
(-fork,)Tj
/F7 1 Tf
2.6695 0 TD
(a)Tj
/F14 1 Tf
6.9738 0 0 6.9738 405.0809 450.132 Tm
(q)Tj
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[(,)-333.3(for)]TJ
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[(Consequen)27.9(tly)83.4(,)]TJ
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[(.)-762.9(Hence,)]TJ
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[(.)-464.3(Hence,)]TJ
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[(.)-464.3(But)]TJ
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[(.)-482.2(Similarly)83.4(,)]TJ
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[(,s)465.2(o)]TJ
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[(.)-761.2(W)83.4(e)-439(rst)-438.8(sho)27.7(w)-438.9(that)]TJ
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[(.F)844.4(o)761.1(r)]TJ
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[(,i)465.3(f)]TJ
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[(,)-298(then)]TJ
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[(.)-429.7(Let)]TJ
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( >
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>>
>>
endobj
55 0 obj
<<
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>>
stream
BT
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0 g
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[(10)-8593.1(G.)-384.8(GR)]TJ
13.1746 0.2527 TD
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[(ATZER,)-384.4(R.)-189.8(W.)-384.7(QUA)31(CKENBUSH,)-384.2(AND)-384.7(E.)-189.7(T.)-384.8(SCHMIDT)]TJ
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[(Fo)-83.4(r)]TJ
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[(,)-474.4(the)-446.1(elemen)27.8(t)]TJ
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(v)Tj
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[(is)-446.1(doubly)-446.1(irreducible)-446.1(in)]TJ
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[(.)-782.9(Therefore,)-474.3(eac)27.8(h)]TJ
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[(is)-444.3(a)-444.3(sublattice)-444.3(of)]TJ
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[(.)-777.3(Let)]TJ
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[(b)-27.7(e)-444.3(the)-444.3(direct)-444.3(pro)-27.8(duct)-444.2(of)-444.2(the)]TJ
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[(Ob)27.8(viously)83.3(,)]TJ
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[(;)-314.8(in)-305.5(other)-305.5(w)27.7(ords,)-311(w)27.7(e)-305.5(can)-305.6(assume)-305.4(that)-305.5()]TJ
/F10 1 Tf
18.0369 0 TD
()Tj
/F6 1 Tf
1.0555 0 TD
[(.)-435.2(Let)]TJ
/F7 1 Tf
3.1991 0 TD
(B)Tj
/F6 1 Tf
-35.1913 -1.2 TD
[(b)-27.7(e)-304.2(an)-304.2(arbitrary)-304.2()-304.1(blo)-27.9(c)27.8(k)0(.)-434.7(Since)-304.2()]TJ
/F10 1 Tf
14.6584 0 TD
()Tj
/F6 1 Tf
1.0556 0 TD
[(,)-310(there)-304.2(is)-304.2(a)-304.2(\(unique\))-304.2()-304.2(blo)-27.8(c)27.7(k)]TJ
ET
415.29 650.529 8.057 -0.399 re
f
BT
/F7 1 Tf
9.9626 0 0 9.9626 415.2767 642.0772 Tm
(B)Tj
/F6 1 Tf
1.1129 0 TD
[(con)27.8(taining)]TJ
/F7 1 Tf
4.832 0 TD
(B)Tj
/F6 1 Tf
0.8087 0 TD
(.)Tj
-35.7222 -1.2333 TD
(Since)Tj
/F7 1 Tf
2.6111 0 TD
(L)Tj
/F6 1 Tf
1.0139 0 TD
[(is)-333.3(isoform,)]TJ
/F7 1 Tf
4.8194 0 TD
(A)Tj
/F10 1 Tf
1.0278 0.2669 TD
()Tj
/F6 1 Tf
0 -0.3169 TD
(=)Tj
/F7 1 Tf
1.0555 0.05 TD
(B)Tj
/F6 1 Tf
1.142 0 TD
(and)Tj
/F7 1 Tf
1.9445 0 TD
(A)Tj
/F10 1 Tf
1.0278 0.2669 TD
()Tj
/F6 1 Tf
0 -0.3169 TD
(=)Tj
ET
283.05 638.169 8.057 -0.399 re
f
BT
/F7 1 Tf
9.9626 0 0 9.9626 283.0615 629.7903 Tm
(B)Tj
/F6 1 Tf
0.8087 0 TD
[(;)-333.3(b)27.8(y)-333.3(niteness,)]TJ
/F7 1 Tf
6.6222 0 TD
(B)Tj
/F6 1 Tf
1.0865 0 TD
(=)Tj
ET
378.45 638.169 8.057 -0.399 re
f
BT
/F7 1 Tf
9.9626 0 0 9.9626 378.4329 629.7903 Tm
(B)Tj
/F6 1 Tf
0.8087 0 TD
[(,)-333.3(that)-333.3(is,)-333.3()-277.8(=)-277.7(.)]TJ
/F12 1 Tf
9.1431 0 TD
()Tj
/F1 1 Tf
-35.2222 -2.1467 TD
[(Corollary)96(.)]TJ
/F8 1 Tf
5.4562 0 TD
[(The)-357.7(lattic)51.1(e)]TJ
/F7 1 Tf
5.0088 0 TD
(L)Tj
/F8 1 Tf
1.0384 0 TD
[(of)-357.7(The)51.2(or)51.1(em)-357.7(1)-357.8(i)0(s)-357.8(r)51.2(e)51.1(gular.)]TJ
/F6 1 Tf
-11.5034 -2.2852 TD
(9.2.)Tj
/F1 1 Tf
2.0555 0 TD
[(P)31.9(erm)32(utable)-539.4(congruences.)]TJ
/F6 1 Tf
13.2409 0 TD
[(The)-469(represen)27.8(tation)-469(result)-469.1(for)-469(p)-27.8(erm)27.9(utable)-469(con-)]TJ
-15.2964 -1.2 TD
[(gruences)-361.4(app)-27.8(eared)-361.6(in)-361.5(G.)-361.6(Gr)]TJ
12.2239 0 TD
[(atzer)-361.5(and)-361.5(E.)-166.7(T.)-361.6(Sc)27.8(hmidt)-361.5([2].)-529(Since)-361.5(sectionally)-361.6(comple-)]TJ
-12.2239 -1.2 TD
[(men)27.9(ted)-377.4(lattices)-377.3(ha)27.8(v)27.8(e)-377.5(p)-27.7(erm)27.8(utable)-377.3(congruences,)-388.3(the)-377.4(stronger)-377.4(congruence-preserving)]TJ
T*
[(extension)-311(v)27.8(ersion)-311(can)-311(b)-27.8(e)-311.1(found)-311(in)-311.1(G.)-311(Gr)]TJ
17.4022 0 TD
[(atzer)-311(and)-311.1(E.)-166.7(T.)-311(Sc)27.8(hmidt)-311.1([4].)-436.9(W)83.3(e)-311.1(w)27.8(ould)-311(lik)27.7(e)]TJ
-17.4022 -1.2 TD
[(to)-333.3(p)-27.7(oin)27.7(t)-333.3(out,)-333.4(that)-333.2(Theorem)-333.3(1)-333.4(con)27.8(tains)-333.3(these)-333.3(statemen)27.8(ts.)-444.5(This)-333.3(follo)27.7(ws)-333.3(from)]TJ
/F1 1 Tf
0 -1.5314 TD
[(Lemma)-383.2(9.)]TJ
/F8 1 Tf
5.4721 0 TD
[(The)-357.8(lattic)51.2(e)]TJ
/F7 1 Tf
5.0089 0 TD
(L)Tj
/F8 1 Tf
1.0383 0 TD
[(of)-357.8(The)51.2(or)51.1(em)-357.7(1)-357.8(i)0(s)-357.7(c)51.1(ongruenc)51.1(e)-357.7(p)51.1(ermutable.)]TJ
-11.5193 -2.1466 TD
[(Pr)51.2(o)51.1(of.)]TJ
/F6 1 Tf
3.1333 0 TD
[(This)-336.9(is)-337(ob)27.9(vious)-336.9(since)-336.9(the)-337(congruences)-336.9(of)]TJ
/F7 1 Tf
17.8058 0 TD
(L)Tj
/F6 1 Tf
1.0175 0 TD
[(are)-336.9(congruences)-336.9(of)]TJ
/F7 1 Tf
8.383 0 TD
(S)Tj
/F6 1 Tf
0.6708 0 TD
[(,)-337.9(whic)27.8(h)-336.9(i)0(s)-336.9(a)]TJ
-31.0104 -1.2 TD
[(direct)-333.3(pro)-27.8(duct)-333.3(of)-333.2(simple)-333.3(latices.)]TJ
/F12 1 Tf
35.2222 0 TD
()Tj
/F6 1 Tf
-35.2222 -2.1466 TD
(9.3.)Tj
/F1 1 Tf
2.0555 0 TD
[(Deterministic)-528.3(isoform)-528.4(lattices.)]TJ
/F6 1 Tf
15.9912 0 TD
(Let)Tj
/F7 1 Tf
1.9179 0 TD
(L)Tj
/F6 1 Tf
1.14 0 TD
[(b)-27.8(e)-459.5(an)-459.5(isoform)-459.5(lattice,)-491(let)-459.5()-459.6(b)-27.7(e)-459.5(a)]TJ
-21.1046 -1.2 TD
[(congruence)-436.5(of)]TJ
/F7 1 Tf
6.515 0 TD
(L)Tj
/F6 1 Tf
0.6805 0 TD
[(,)-462.5(and)-436.7(let)]TJ
/F7 1 Tf
4.3358 0 TD
(A)Tj
/F9 1 Tf
6.9738 0 0 6.9738 249.0271 454.3831 Tm
()Tj
/F6 1 Tf
9.9626 0 0 9.9626 259.5626 455.8775 Tm
[(b)-27.8(e)-436.7(a)-436.6(congruence)-436.6(class)-436.7(of)-436.6(.)-754.5(Then)-436.6(an)27.8(y)-436.6(congruence)]TJ
-13.3388 -1.2 TD
[(class)-345.8(of)-345.7()-345.8(i)0(s)-345.8(isomorphic)-345.8(to)]TJ
/F7 1 Tf
11.9053 0 TD
(A)Tj
/F9 1 Tf
6.9738 0 0 6.9738 252.7532 442.4279 Tm
()Tj
/F6 1 Tf
9.9626 0 0 9.9626 258.938 443.9223 Tm
[(.)-481.8(H)0(o)27.7(w)27.8(ev)27.7(er,)-348.9(if)-345.8()-345.8(and)-345.8()-345.8(are)-345.8(congruences)-345.8(of)]TJ
/F7 1 Tf
18.5708 0 TD
(L)Tj
/F6 1 Tf
0.6806 0 TD
0.3473 Tc
[(,i)347.3(tm)347.3(a)375(y)]TJ
-32.5275 -1.2 TD
0 Tc
[(happ)-27.7(en)-368(that)]TJ
/F7 1 Tf
5.7637 0 TD
(A)Tj
/F9 1 Tf
6.9738 0 0 6.9738 191.5666 430.4727 Tm
()Tj
/F6 1 Tf
9.9626 0 0 9.9626 201.4177 431.9671 Tm
(and)Tj
/F7 1 Tf
1.9791 0 TD
(A)Tj
/F9 1 Tf
6.9738 0 0 6.9738 228.6067 430.4727 Tm
()Tj
/F6 1 Tf
9.9626 0 0 9.9626 238.8862 431.9671 Tm
[(as)-367.9(lattices)-368(are)-367.9(isomorphic.)-548.3(Let)-368(us)-368(call)-367.9(an)-368(isoform)-367.9(lattice)]TJ
/F7 1 Tf
-11.2634 -1.2 TD
(L)Tj
/F8 1 Tf
1.0645 0 TD
(deterministic)Tj
/F6 1 Tf
5.6861 0 TD
[(,)-396.7(i)0(f)-384(this)-384(cannot)-384(happ)-27.7(en;)-409.3(in)-384.1(other)-384(w)27.8(ords,)-396.6(if)-384.1()]TJ
/F10 1 Tf
19.9968 0 TD
()Tj
/F6 1 Tf
T*
[(=)-362.3()-384(are)-384(congruences)]TJ
-26.7474 -1.2 TD
(of)Tj
/F7 1 Tf
1.1388 0 TD
(L)Tj
/F6 1 Tf
0.6806 0 TD
[(,)-333.3(then)]TJ
/F7 1 Tf
2.8889 0 TD
(A)Tj
/F9 1 Tf
6.9738 0 0 6.9738 181.052 406.5624 Tm
()Tj
/F6 1 Tf
9.9626 0 0 9.9626 190.5584 408.0568 Tm
(and)Tj
/F7 1 Tf
1.9444 0 TD
(A)Tj
/F9 1 Tf
6.9738 0 0 6.9738 217.4017 406.5624 Tm
()Tj
/F6 1 Tf
9.9626 0 0 9.9626 227.3365 408.0568 Tm
[(are)-333.4(not)-333.3(isomorphic.)]TJ
/F1 1 Tf
-10.1041 -1.5313 TD
[(Lemma)-383.2(10.)]TJ
/F8 1 Tf
6.0471 0 TD
[(The)-357.8(lattic)51.2(e)]TJ
/F7 1 Tf
5.0089 0 TD
(L)Tj
/F8 1 Tf
1.0383 0 TD
[(of)-357.8(The)51.2(or)51.1(em)-357.7(1)-357.8(c)51.1(an)-357.8(b)51.1(e)-357.7(c)51.1(onstructe)51.1(d)-357.8(t)0(o)-357.8(b)51.2(e)-357.8(deterministic.)]TJ
-12.0943 -2.1467 TD
[(Pr)51.2(o)51.1(of.)]TJ
/F6 1 Tf
3.1333 0 TD
[(The)-279.7(size)]TJ
/F10 1 Tf
3.8427 0 TD
(|)Tj
/F7 1 Tf
0.2778 0 TD
(A)Tj
/F9 1 Tf
6.9738 0 0 6.9738 206.4119 369.9198 Tm
()Tj
/F10 1 Tf
9.9626 0 0 9.9626 212.5967 371.4142 Tm
(|)Tj
/F6 1 Tf
0.5575 0 TD
[(is)-279.7(the)-279.7(pro)-27.8(duct)-279.7(of)-279.6(the)]TJ
/F10 1 Tf
9.0735 0 TD
(|)Tj
/F7 1 Tf
0.2778 0 TD
(S)Tj
/F9 1 Tf
6.9738 0 0 6.9738 317.4236 369.9198 Tm
()Tj
/F10 1 Tf
9.9626 0 0 9.9626 323.6085 371.4142 Tm
(|)Tj
/F6 1 Tf
0.2778 0 TD
0.2904 Tc
(,)Tj
/F10 1 Tf
1.5682 0 TD
0 Tc
()Tj
/F6 1 Tf
0.9445 0 TD
(M\(Con)Tj
/F7 1 Tf
3.25 0 TD
(K)Tj
/F6 1 Tf
0.9208 0 TD
[(\).)-426.5(Since)-279.7(w)27.7(e)-279.7(can)-279.7(easily)]TJ
-26.7287 -1.2 TD
[(construct)-315(the)]TJ
/F7 1 Tf
6.0829 0 TD
(S)Tj
/F9 1 Tf
6.9738 0 0 6.9738 193.3838 357.9647 Tm
()Tj
/F6 1 Tf
9.9626 0 0 9.9626 199.5686 359.459 Tm
[(-s)-315.1(so)-315.1(that)-315.1(all)]TJ
/F10 1 Tf
5.7715 0 TD
(|)Tj
/F7 1 Tf
0.2778 0 TD
(S)Tj
/F9 1 Tf
6.9738 0 0 6.9738 265.9447 357.9647 Tm
()Tj
/F10 1 Tf
9.9626 0 0 9.9626 272.1295 359.459 Tm
(|)Tj
/F6 1 Tf
0.2778 0 TD
[(-s)-315(are)-315.1(distinct)-315.1(primes,)-318.7(the)-315.1(statemen)27.8(t)-315.1(follo)27.8(ws.)]TJ
/F12 1 Tf
20.3442 0 TD
()Tj
/F6 1 Tf
-35.2222 -2.1466 TD
(9.4.)Tj
/F1 1 Tf
2.0555 0 TD
[(A)-372.8(small)-372.7(construction.)]TJ
/F6 1 Tf
11.2742 0 TD
(Let)Tj
/F7 1 Tf
1.7825 0 TD
(K)Tj
/F6 1 Tf
1.245 0 TD
[(b)-27.8(e)-324.1(a)-324.2(nite)-324.2(lattice.)-441.4(F)83.4(o)0(r)-324.1(e)0(v)27.7(ery)-324.1(meet-irreducible)]TJ
-16.3572 -1.2 TD
[(congruence)-404()-404.1(o)0(f)]TJ
/F7 1 Tf
7.5762 0 TD
(K)Tj
/F6 1 Tf
0.9208 0 TD
[(,)-421.8(w)27.7(e)-404.1(form)-404.1(the)-404.1(quotien)27.7(t)-404.1(lattice)]TJ
/F7 1 Tf
13.6119 0 TD
0.016 Tc
(K/)Tj
/F6 1 Tf
1.3653 0 TD
0 Tc
[(,)-421.8(and)-404.1(extend)-404.1(it)-404.1(to)-404.1(a)-404.1(nite,)]TJ
-23.4742 -1.2 TD
[(simple,)-377.3(separable)-368.5(lattice)]TJ
/F7 1 Tf
10.9615 0 TD
(S)Tj
/F9 1 Tf
6.9738 0 0 6.9738 241.9875 312.6685 Tm
()Tj
/F6 1 Tf
9.9626 0 0 9.9626 248.1733 314.1629 Tm
[(.)-549.9(W)83.3(e)-368.5(can)-368.5(construct)]TJ
/F7 1 Tf
8.8861 0 TD
(S)Tj
/F9 1 Tf
6.9738 0 0 6.9738 342.8105 312.6685 Tm
()Tj
/F6 1 Tf
9.9626 0 0 9.9626 352.6675 314.1629 Tm
[(b)27.8(y)-368.5(adding)-368.4(only)-368.6(three)-368.5(elemen)27.8(ts)]TJ
-22.6842 -1.2 TD
(to)Tj
/F7 1 Tf
1.1692 0 TD
0.016 Tc
(K/)Tj
/F6 1 Tf
1.3653 0 TD
0 Tc
[()-280.4(a)0(s)-280.5(follo)27.8(ws:)]TJ
/F7 1 Tf
5.8232 0 TD
0.016 Tc
(K/)Tj
/F6 1 Tf
1.3653 0 TD
0 Tc
[()-280.4(i)0(s)-280.4(a)-280.4(sub)-27.7(directly)-280.4(irreducible)-280.3(lattice;)-298.1(so)-280.4(it)-280.4(has)-280.4(a)-280.3(c)0(o)27.8(v)27.8(ering)-280.3(pair)]TJ
/F7 1 Tf
-9.723 -1.2 TD
(a)Tj
/F10 1 Tf
0.8578 0 TD
()Tj
/F7 1 Tf
1.107 0 TD
(b)Tj
/F6 1 Tf
0.7934 0 TD
[(suc)27.8(h)-364.2(that)-364.3(Cg)-13.9(\()]TJ
/F7 1 Tf
6.1091 0 TD
[(a,)-166.6(b)]TJ
/F6 1 Tf
1.4022 0 TD
[(\))-364.2(i)0(s)-364.2(the)-364.2(unique)-364.2(minimal)-364.2(congruence)-364.2(of)]TJ
/F7 1 Tf
17.1135 0 TD
0.0159 Tc
(K/)Tj
/F6 1 Tf
1.3652 0 TD
0 Tc
[(.)-537.1(Dene)-364.2(a)-364.2(new)]TJ
-28.7482 -1.2 TD
[(elemen)27.8(t)]TJ
/F7 1 Tf
3.7463 0 TD
(s)Tj
/F9 1 Tf
6.9738 0 0 6.9738 168.6655 276.803 Tm
()Tj
/F6 1 Tf
9.9626 0 0 9.9626 174.8513 278.2974 Tm
[(,)-398.2(a)-385.2(relativ)27.8(e)-385.2(complemen)27.7(t)-385.2(o)0(f)]TJ
/F7 1 Tf
11.9975 0 TD
(a)Tj
/F6 1 Tf
0.9138 0 TD
[(in)-385.2([0)]TJ
/F7 1 Tf
1.9963 0 TD
0.1667 Tc
(,b)Tj
/F6 1 Tf
0.8736 0 TD
0 Tc
[(],)-398.2(and)-385.2(t)27.8(w)27.7(o)-385.2(others:)]TJ
/F7 1 Tf
8.3919 0 TD
(v)Tj
/F9 1 Tf
6.9738 0 0 6.9738 420.8658 281.9128 Tm
(1)Tj
-0.0512 -0.9119 TD
()Tj
/F6 1 Tf
9.9626 0 0 9.9626 430.5315 278.2974 Tm
(and)Tj
/F7 1 Tf
1.9964 0 TD
(v)Tj
/F9 1 Tf
6.9738 0 0 6.9738 455.6075 281.9128 Tm
(2)Tj
-0.0513 -0.9119 TD
()Tj
/F6 1 Tf
9.9626 0 0 9.9626 461.4346 278.2974 Tm
[(,)-398.2(new)]TJ
-33.6017 -1.2 TD
[(separator)-333.3(elemen)27.8(ts.)-444.4(The)-333.4(resulting)-333.3(lattice)-333.2(is)-333.4(simple.)]TJ
0 -2.2853 TD
(9.5.)Tj
/F1 1 Tf
2.0555 0 TD
[(Rectangular)-439(vs.)-678.6(cubic)-439.1(extensions.)]TJ
/F6 1 Tf
17.6753 0 TD
[(In)-381.9(G.)-381.9(Gr)]TJ
3.9193 0 TD
[(atzer)-381.9(and)-381.9(E.)-166.7(T.)-381.9(Sc)27.8(hmidt)-381.9([3],)]TJ
-23.6501 -1.2 TD
[(w)27.8(e)-486.8(in)27.7(tro)-27.7(duced)-486.7(a)]TJ
/F8 1 Tf
7.7686 0 TD
[(r)51.2(e)51.1(ctangular)-498.8(extension)]TJ
/F13 1 Tf
9.9595 0 TD
(R)Tj
/F6 1 Tf
0.7222 0 TD
(\()Tj
/F7 1 Tf
0.3889 0 TD
(K)Tj
/F6 1 Tf
0.9209 0 TD
[(\))-486.7(o)0(f)-486.7(a)-486.8(nite)-486.8(lattice)]TJ
/F7 1 Tf
9.0726 0 TD
(K)Tj
/F6 1 Tf
1.4076 0 TD
[(as)-486.7(the)-486.8(direct)]TJ
-30.2403 -1.2 TD
[(pro)-27.8(duct)-296.5(of)-296.6(all)-296.6(sub)-27.7(direct)-296.6(quotien)27.8(ts)]TJ
/F7 1 Tf
14.8053 0 TD
0.016 Tc
(K/)Tj
/F6 1 Tf
1.3653 0 TD
0 Tc
[(.)-432.2(W)83.4(e)-296.7(then)-296.5(selected)]TJ
/F7 1 Tf
9.0498 0 TD
(S)Tj
/F6 1 Tf
0.6708 0 TD
(\()Tj
/F7 1 Tf
0.3889 0 TD
0.016 Tc
(K/)Tj
/F6 1 Tf
1.3653 0 TD
0 Tc
[(\),)-304(a)-296.6(nite,)-303.9(simple,)]TJ
-27.6454 -1.2 TD
[(sectionally)-333.3(complemen)27.8(ted)-333.3(extension)-333.4(of)]TJ
/F7 1 Tf
17.0666 0 TD
0.016 Tc
(K/)Tj
/F6 1 Tf
1.3653 0 TD
0 Tc
[(,)-333.3(and)-333.3(formed)]TJ
/F11 1 Tf
-8.8569 -1.8043 TD
(\012)Tj
/F13 1 Tf
-0.0834 -0.2583 TD
(R)Tj
/F6 1 Tf
0.7223 0 TD
(\()Tj
/F7 1 Tf
0.3888 0 TD
(K)Tj
/F6 1 Tf
0.9209 0 TD
0.2777 Tc
(\)=)Tj
/F11 1 Tf
1.7222 0.95 TD
0 Tc
()Tj
/F6 1 Tf
1.2778 -0.95 TD
(\()Tj
/F7 1 Tf
0.5555 0 TD
(S)Tj
/F6 1 Tf
0.6709 0 TD
(\()Tj
/F7 1 Tf
0.3888 0 TD
0.016 Tc
(K/)Tj
/F6 1 Tf
1.3653 0 TD
0 Tc
(\))Tj
/F10 1 Tf
1.3889 0 TD
(|)Tj
/F6 1 Tf
0.5556 0 TD
()Tj
/F10 1 Tf
1 0 TD
()Tj
/F6 1 Tf
0.9444 0 TD
(M\(Con)Tj
/F7 1 Tf
3.25 0 TD
(K)Tj
/F6 1 Tf
0.9208 0 TD
0.1667 Tc
(\)\))Tj
-25.5638 -2.0627 TD
0 Tc
(of)Tj
/F13 1 Tf
1.1388 0 TD
(R)Tj
/F6 1 Tf
0.7223 0 TD
(\()Tj
/F7 1 Tf
0.3889 0 TD
(K)Tj
/F6 1 Tf
0.9208 0 TD
(\).)Tj
-1.9708 -1.2 TD
[(W)83.4(e)-455.6(c)27.7(hanged)-455.5(the)-455.6(terminology)-455.5(for)-455.5(v)55.5(arious)-455.5(reasons.)-811(\(i\))-455.5(It)-455.6(seems)-455.5(immaterial,)-486.1(in)]TJ
-1.2 -1.2 TD
[(general,)-387.9(that)]TJ
/F7 1 Tf
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(p)Tj
/F6 1 Tf
9.9626 0 0 9.9626 151.8246 654.0323 Tm
(=)Tj
/F7 1 Tf
1.1389 0 TD
(v)Tj
/F6 1 Tf
0.5206 0 TD
[(.)-594.4(W)83.4(e)-383.4(n)0(o)27.9(w)-383.4(v)27.8(erify)-383.3(that)]TJ
/F1 1 Tf
9.8249 0 TD
(a)Tj
/F7 1 Tf
0.9201 0 TD
(<)Tj
/F1 1 Tf
1.1388 0 TD
(c)Tj
/F6 1 Tf
0.5111 0 TD
[(.)-594.4(Let)]TJ
/F7 1 Tf
2.7139 0 TD
0.3611 Tc
(p