%!PS-Adobe-2.0 %%Creator: dvipsk 5.58f Copyright 1986, 1994 Radical Eye Software %%Title: algage.dvi %%Pages: 8 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%EndComments %DVIPSCommandLine: dvips -o algage.ps algage.dvi %DVIPSParameters: dpi=300, compressed, comments removed %DVIPSSource: TeX output 1999.09.21:1438 %%BeginProcSet: texc.pro /TeXDict 250 dict def TeXDict begin /N{def}def /B{bind def}N /S{exch}N /X{S N}B /TR{translate}N /isls false N /vsize 11 72 mul N /hsize 8.5 72 mul N /landplus90{false}def /@rigin{isls{[0 landplus90{1 -1}{-1 1} ifelse 0 0 0]concat}if 72 Resolution div 72 VResolution div neg scale isls{landplus90{VResolution 72 div vsize mul 0 exch}{Resolution -72 div hsize mul 0}ifelse TR}if Resolution VResolution vsize -72 div 1 add mul TR[matrix currentmatrix{dup dup round sub abs 0.00001 lt{round}if} forall round exch round exch]setmatrix}N /@landscape{/isls true N}B /@manualfeed{statusdict /manualfeed true put}B /@copies{/#copies X}B /FMat[1 0 0 -1 0 0]N /FBB[0 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b Fo(on)14 b(\012,)g(then)h Fi(A)p Fo(\()p Fi(C)s Fo(\))e(is)i(a)f(subalgebra)h(of)238 1211 y(the)g(reduced)g(incidence)i(algebra)d Fi(A)h Fo(on)f(\012)g (\(and)g(this)h(is)f(equiv)m(alen)o(t)i(to)e Fi(C)j Fo(ha)o(ving)d(the) 238 1268 y Ff(joint)e(emb)n(e)n(dding)f(pr)n(op)n(erty)p Fo(,)h(that)e(is,)h(an)o(y)g(t)o(w)o(o)e(mem)o(b)q(ers)i(of)f Fi(C)j Fo(can)e(b)q(e)g(sim)o(ultaneously)238 1324 y(em)o(b)q(edded)17 b(in)f(a)f(mem)o(b)q(er)g(of)g Fi(C)s Fo(\).)193 1431 y Fi(\017)22 b Fo(If)11 b Fi(C)h Fo(is)f(the)f(age)g(of)g(a)g (homogeneous)g(relational)h(structure)f Fk(M)15 b Fo(on)10 b(\012,)h(then)f Fi(A)p Fo(\()p Fi(C)s Fo(\))i(=)h Fi(A)1712 1415 y Fl(G)1742 1431 y Fo(,)238 1487 y(where)20 b Fk(G)f Fo(=)h(Aut\()p Fk(M)5 b Fo(\))19 b(\(and)g(this)h(is)g(equiv)m(alen)o (t)h(to)e Fi(C)j Fo(ha)o(ving)d(the)h Ff(amalgamation)238 1544 y(pr)n(op)n(erty)p Fo(,)h(that)e(is,)i(an)o(y)e(amalgam)g(of)g(t)o (w)o(o)f(mem)o(b)q(ers)i(of)f Fi(C)j Fo(with)e(a)g(common)f(sub-)238 1600 y(structure)c(can)g(b)q(e)h(em)o(b)q(edded)h(in)f(a)f(mem)o(b)q (er)g(of)g Fi(C)s Fo(\).)125 1704 y(See,)g(for)g(example,)g(Cameron)g ([3)o(])g(for)f(discussion.)125 1862 y Fp(2)66 b(P)n(olynomial)25 b(algebras)125 1967 y Fo(There)13 b(are)g(only)h(t)o(w)o(o)e(tec)o (hniques)j(I)f(kno)o(w)f(for)f(determining)j(the)f(structure)f(of)g (the)g(algebras)125 2023 y Fi(A)161 2007 y Fl(G)204 2023 y Fo(or)g Fi(A)p Fo(\()p Fi(C)s Fo(\).)19 b(The)14 b(\014rst)f(is)i (based)f(on)g(the)g(simple)h(observ)m(ation)f(that,)f(regarding)h Fk(G)7 b Fi(\002)g Fk(H)18 b Fo(as)125 2080 y(a)e(p)q(erm)o(utation)h (group)f(on)h(the)g(disjoin)o(t)g(union)h(of)e(the)h(sets)g(on)f(whic)o (h)i Fk(G)e Fo(and)h Fk(H)j Fo(act,)d(w)o(e)125 2136 y(ha)o(v)o(e)732 2202 y Fi(A)768 2183 y Fl(G)p Fh(\002)p Fl(H)870 2202 y Fo(=)c Fi(A)954 2183 y Fl(G)994 2202 y Fi(\012)1029 2209 y Fd(Q)1064 2202 y Fi(A)1100 2183 y Fl(H)1134 2202 y Fk(:)125 2297 y Fo(Let)g Fk(S)i Fo(denote)e(the)g 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b(example,)i(if)g Fk(H)i Fo(is)d(the)g(symmetric)g(group)f Fk(S)1718 2643 y Fl(n)1742 2636 y Fo(,)125 2692 y(then)e Fi(A)265 2676 y Fl(S)r Fm(W)m(r)q Fl(S)357 2680 y Fg(n)397 2692 y Fo(is)h(the)f(p)q(olynomial) i(algebra)e(generated)h(b)o(y)f(the)g Fk(n)h Fo(elemen)o(tary)f (symmetric)125 2749 y(functions,)e(b)o(y)g(Newton's)g(Theorem.)19 b(\(Note)14 b(that)f Fi(A)1061 2732 y Fl(S)r Fm(W)m(r)q Fl(H)1180 2749 y Fo(is)h(alw)o(a)o(ys)g(an)g(in)o(tegral)g(domain,)125 2805 y(but)h(almost)g(nev)o(er)g(a)g(p)q(olynomial)i(algebra.\))p eop %%Page: 3 3 3 2 bop 140 162 a Fe(The)15 b(algebra)g(of)g(an)g(age)1159 b Fo(3)191 311 y(In)16 b(this)g(case,)g(the)f(n)o(um)o(b)q(ers)h Fk(f)731 318 y Fl(n)755 311 y Fo(\()p Fk(S)c Fo(W)l(r)e Fk(H)t Fo(\))15 b(can)h(b)q(e)g(calculated)h(b)o(y)f(Molien's)g (Theorem,)125 368 y(whic)o(h)k(turns)g(out)f(to)g(b)q(e)h(a)g(sp)q (ecial)h(case)f(of)f(a)h(\\cycle)g(index)h(theory")e(for)g (oligomorphic)125 424 y(p)q(erm)o(utation)c(groups)g(\(see)g([3)o(]\).) 191 535 y(The)e(second)h(approac)o(h)f(requires)h(that)e(the)h(class)h Fi(C)i Fo(has)d(a)f(\\go)q(o)q(d)h(notion)h(of)e(connected-)125 591 y(ness",)j(as)h(follo)o(ws.)22 b(I)16 b(will)h(giv)o(e)f(an)g (axiomatic)g(treatmen)o(t,)f(since)i(in)f(one)g(of)g(the)g(examples)125 648 y(b)q(elo)o(w,)21 b(w)o(ords)f(lik)o(e)h(\\connected")g(and)f(\\in) o(v)o(olv)o(emen)o(t")g(ha)o(v)o(e)g(meanings)h(quite)g(di\013eren)o(t) 125 704 y(from)14 b(their)i(usual)g(ones.)k(W)l(e)15 b(require)193 801 y Fi(\017)22 b Fo(a)15 b(distinguished)j(sub)q(class) e(of)f Fi(C)j Fo(consisting)e(of)e(\\connected")i(structures;)193 899 y Fi(\017)22 b Fo(a)c(partial)g(order)f Fi(\024)h Fo(called)h(\\in)o(v)o(olv)o(emen)o(t")f(on)g(the)f(class)i(of)e Fk(n)p Fo(-elemen)o(t)i(structures)238 955 y(for)c(eac)o(h)g Fk(n)p Fo(;)193 1053 y Fi(\017)22 b Fo(a)17 b(binary)l(,)g(comm)o (utativ)o(e)f(and)h(asso)q(ciativ)o(e)g(\\comp)q(osition")g Fi(\016)g Fo(suc)o(h)g(that,)f(if)h Fk(X)j Fo(and)238 1110 y Fk(Y)25 b Fo(are)15 b(structures)f(with)h Fk(n)g Fo(and)g Fk(m)f Fo(p)q(oin)o(ts)h(resp)q(ectiv)o(ely)l(,)h(then)f Fk(X)d Fi(\016)d Fk(Y)25 b Fo(is)15 b(a)g(structure)238 1166 y(with)h Fk(n)10 b Fo(+)h Fk(m)k Fo(p)q(oin)o(ts.)125 1263 y(Assume)g(that)f(the)i(follo)o(wing)g(conditions)g(hold:)147 1360 y(A1)g(Let)h Fk(S)i Fo(b)q(e)e(a)f(structure)g(whic)o(h)h(is)g (partitioned)g(in)o(to)f(disjoin)o(t)h(induced)h(substructures)238 1416 y Fk(S)266 1423 y Fm(1)286 1416 y Fk(;)8 b(S)335 1423 y Fm(2)354 1416 y Fk(;)g(:)g(:)g(:)k Fo(.)20 b(Then)15 b Fk(S)621 1423 y Fm(1)651 1416 y Fi(\016)10 b Fk(S)712 1423 y Fm(2)742 1416 y Fi(\016)f Fk(:)f(:)g(:)j Fi(\024)i Fk(S)s Fo(.)147 1514 y(A2)23 b(An)o(y)f(structure)h(has)g(a)f(unique)i (represen)o(tation)f(as)f(a)h(comp)q(osition)g(of)f(connected)238 1571 y(structures.)125 1681 y Fc(Theorem)17 b(2.1)22 b Ff(If)15 b(al)r(l)h(the)g(ab)n(ove)f(c)n(onditions)g(hold,)h(then)f Fi(A)p Fo(\()p Fi(C)s Fo(\))g Ff(is)h(a)g(p)n(olynomial)f(algebr)n(a,) 125 1737 y(gener)n(ate)n(d)g(by)h(the)h(char)n(acteristic)f(functions)g (of)g(the)h(c)n(onne)n(cte)n(d)d(structur)n(es.)125 1848 y(Pr)n(o)n(of.)21 b Fo(If)c Fi(j)p Fk(S)s Fi(j)12 b Fo(=)i Fk(n)p Fo(,)i(then)g Fk(S)j Fo(is)d(a)g(disjoin)o(t)g(union)h Fk(S)1040 1855 y Fm(1)1070 1848 y Fi([)11 b Fk(S)1139 1855 y Fm(2)1169 1848 y Fi([)g Fk(:)d(:)g(:)14 b Fo(of)h(connected)i (structures;)125 1904 y(so)g(w)o(e)g(ha)o(v)o(e)g(a)h(bijection)h(b)q (et)o(w)o(een)e(c)o(haracteristic)h(functions)h Fk(\037)1274 1911 y Fl(S)1317 1904 y Fo(\(the)f(basis)g(elemen)o(ts)g(of)125 1961 y Fk(V)152 1968 y Fl(n)175 1961 y Fo(\()p Fi(C)s Fo(\)\))g(and)j(monomials)f Fk(\036)627 1968 y Fl(S)674 1961 y Fo(=)h Fk(\037)758 1968 y Fl(S)779 1973 y Fb(1)799 1961 y Fk(\037)827 1968 y Fl(S)848 1973 y Fb(2)876 1961 y Fk(:)8 b(:)g(:)18 b Fo(of)i(total)g(w)o(eigh)o(t)g Fk(n)p Fo(.)35 b(Consider)20 b(the)h(matrix)125 2017 y(expressing)15 b(the)f(monomials)h Fk(\036)674 2024 y Fl(S)714 2017 y Fo(in)g(terms)f(of)g(the)g(basis)h(elemen)o(ts)g Fk(\037)1344 2025 y Fl(S)1367 2015 y Fa(0)1381 2017 y Fo(.)20 b(The)15 b(co)q(e\016cien)o(t)g(of)125 2074 y Fk(\037)153 2081 y Fl(S)192 2074 y Fo(in)g(the)e(ro)o(w)g(corresp)q (onding)i(to)e Fk(\036)778 2081 y Fl(S)817 2074 y Fo(is)i(non-zero.)k (Supp)q(ose)c(that)e Fk(\037)1365 2081 y Fl(S)1388 2072 y Fa(0)1416 2074 y Fo(also)h(has)f(non-zero)125 2130 y(co)q(e\016cien)o(t.)28 b(Then)18 b Fk(S)514 2114 y Fh(0)543 2130 y Fo(can)g(b)q(e)g(partitioned)h(in)o(to)f(induced)h (substructures)f(isomorphic)h(to)125 2187 y Fk(S)153 2194 y Fm(1)172 2187 y Fk(;)8 b(S)221 2194 y Fm(2)240 2187 y Fk(;)g(:)g(:)g(:)m Fo(;)21 b(so)d Fk(S)k Fo(=)d Fk(S)538 2194 y Fm(1)570 2187 y Fi(\016)13 b Fk(S)634 2194 y Fm(2)666 2187 y Fi(\016)f Fk(:)c(:)g(:)17 b Fi(\024)i Fk(S)858 2170 y Fh(0)870 2187 y Fo(.)31 b(Th)o(us)19 b(the)g(matrix)f(is)i(upp)q(er)g(triangular)f(with)125 2243 y(non-zero)12 b(diagonal,)i(and)f(hence)g(in)o(v)o(ertible.)21 b(So)13 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Fo(is)h(equal)g(to)f(the)g (n)o(um)o(b)q(er)h(of)e(orbits)h(of)g Fk(G)g Fo(on)g Fk(n)p Fo(-sets.)125 992 y Ff(Example)h(3.)k Fo(Let)15 b Fk(A)h Fo(b)q(e)f(a)g(\014xed)h(alphab)q(et)g(of)f(\014nite)h(size)g Fk(q)r Fo(,)e(and)h(let)h Fi(C)f Fo(=)e Fk(A)1472 976 y Fh(\003)1507 992 y Fo(b)q(e)j(the)f(set)g(of)125 1049 y(w)o(ords)e(in)h Fk(A)p Fo(.)20 b(\(Here)13 b(a)h(w)o(ord)f(of)g (length)i Fk(n)f Fo(is)g(regarded)g(as)f(an)h Fk(n)p Fo(-set)g(carrying)g(a)f(total)h(order)125 1105 y(and)e Fk(q)h Fo(unary)f(relations)h Fk(R)587 1112 y Fm(1)606 1105 y Fk(;)8 b(:)g(:)g(:)d(;)j(R)743 1112 y Fl(q)761 1105 y Fo(,)k(where)g(eac)o(h)g(elemen)o(t)h(of)e(the)h(set)g (satis\014es)g(exactly)g(one)125 1162 y(of)k(the)g(unary)h(relations;)h (the)e(w)o(ord)g Fk(a)806 1169 y Fm(1)826 1162 y Fk(a)850 1169 y Fm(2)877 1162 y Fk(:)8 b(:)g(:)e(a)962 1169 y Fl(q)998 1162 y Fo(corresp)q(onds)17 b(to)f(the)g Fk(n)p Fo(-set)h Fi(f)p Fk(x)1549 1169 y Fm(1)1569 1162 y Fk(;)8 b(:)g(:)g(:)t(;)g(x)1696 1169 y Fl(n)1719 1162 y Fi(g)p Fo(,)125 1218 y(with)19 b Fk(x)258 1225 y Fm(1)298 1218 y Fk(<)h(x)379 1225 y Fm(2)418 1218 y Fk(<)h(:)8 b(:)g(:)17 b(<)j(x)627 1225 y Fl(n)670 1218 y Fo(and)g(in)g(whic)o(h)h Fk(x)981 1225 y Fl(i)1014 1218 y Fo(satis\014es)f Fk(R)1221 1225 y Fl(a)1240 1230 y Fg(i)1254 1218 y Fo(.\))33 b(The)19 b(algebra)h Fi(A)p Fo(\()p Fk(A)1667 1202 y Fh(\003)1687 1218 y Fo(\))f(is)125 1275 y(the)e Ff(shu\017e)h(algebr)n(a)e Fo(whic)o(h)i(arises)g(in)g(the)f(theory)f(of)h(free)g(Lie)i(algebras)e ([12)o(].)25 b(The)17 b(name)125 1331 y(comes)f(from)g(the)h(fact)f (that)g(the)h(pro)q(duct)g(of)f(t)o(w)o(o)g(w)o(ords)g(is)h(the)g(sum)g (of)f(all)i(w)o(ords)e(whic)o(h)125 1387 y(can)e(b)q(e)i(obtained)f(b)o (y)g(\\sh)o(u\017ing")g(them)f(together,)g(with)h(appropriate)g(m)o (ultiplicities.)23 b(F)l(or)125 1444 y(example,)551 1508 y(\()p Fk(aab)p Fo(\))9 b Fi(\001)g Fo(\()p Fk(ab)p Fo(\))j(=)h Fk(abaab)c Fo(+)i(3)p Fk(aabab)d Fo(+)j(6)p Fk(aaabb:)191 1600 y Fo(Also,)k Fk(A)340 1584 y Fh(\003)374 1600 y Fo(is)g(the)g(age)f(of)h(a)f(homogeneous)h(relational)g(structure)g Fk(M)5 b Fo(\()p Fk(q)r Fo(\))14 b(whic)o(h)h(is)h(order-)125 1657 y(isomorphic)21 b(to)f Fj(Q)d Fo(and)k(in)g(whic)o(h)g(the)g(set)f (of)g(elemen)o(ts)h(satisfying)g(eac)o(h)f(relation)h Fk(R)1690 1664 y Fl(i)1724 1657 y Fo(is)125 1713 y(dense;)f(in)f(other) f(w)o(ords,)g(a)g(partition)h(of)f Fj(Q)d Fo(in)o(to)j Fk(q)j Fo(dense)e(subsets.)29 b(Suc)o(h)19 b(a)f(partition)h(is)125 1770 y(unique)d(up)g(to)e(order-isomorphism)i(of)f Fj(Q)p Fo(.)i(Let)f Fk(G)p Fo(\()p Fk(q)r Fo(\))11 b(=)i(Aut\()p Fk(M)5 b Fo(\()p Fk(q)r Fo(\)\).)191 1826 y(T)l(ak)o(e)16 b(a)g(total)f(order)h(on)g Fk(A)p Fo(,)g(and)g(de\014ne)h(the)g Ff(lexic)n(o)n(gr)n(aphic)f(or)n(der)h Fo(on)f Fk(A)1482 1810 y Fh(\003)1518 1826 y Fo(in)g(the)h(usual)125 1883 y(w)o(a)o(y:)h(that)d(is,)g Fk(a)415 1890 y Fm(1)443 1883 y Fk(:)8 b(:)g(:)d(a)527 1890 y Fl(m)573 1883 y Fk(<)13 b(b)641 1890 y Fm(1)668 1883 y Fk(:)8 b(:)g(:)d(b)748 1890 y Fl(n)787 1883 y Fo(if)15 b(and)h(only)f(if)h Ff(either)193 1984 y Fi(\017)22 b Fk(m)13 b(<)g(n)p Fo(,)i(and)g Fk(a)506 1991 y Fl(i)533 1984 y Fo(=)e Fk(b)601 1991 y Fl(i)630 1984 y Fo(for)h Fk(i)e Fo(=)h(1)p Fk(;)8 b(:)g(:)g(:)d(;)j(m)p Fo(;)14 b Ff(or)193 2088 y Fi(\017)22 b Fo(for)15 b(some)g Fk(l)e(<)g Fo(min)p Fi(f)p Fk(m;)8 b(n)p Fi(g)p Fo(,)14 b(w)o(e)h(ha)o(v)o(e)g Fk(a)929 2095 y Fl(i)956 2088 y Fo(=)e Fk(b)1024 2095 y Fl(i)1052 2088 y Fo(for)i Fk(i)d Fo(=)h(1)p Fk(;)8 b(:)g(:)g(:)d(;)j(l)q Fo(,)13 b(and)i Fk(a)1476 2095 y Fl(l)p Fm(+1)1547 2088 y Fk(<)e(b)1615 2095 y Fl(l)p Fm(+1)1673 2088 y Fo(.)125 2189 y(A)22 b(non-empt)o(y)h(w)o(ord)f Fk(w)j Fi(2)h Fk(A)684 2173 y Fh(\003)726 2189 y Fo(is)d(a)f Ff(Lyndon)g(wor)n(d)i Fo(if,)g(whenev)o(er)f Fk(w)j Fo(=)f Fk(xy)f Fo(with)f Fk(x;)8 b(y)125 2246 y Fo(non-empt)o(y)l(,)24 b(w)o(e)f(ha)o(v)o(e)f Fk(w)k(<)g(y)r Fo(;)g(that)c(is,)j Fk(w)e Fo(is)h(less)f(than)g(an)o(y) f(prop)q(er)h(cyclic)i(shift)e(of)125 2302 y(itself.)40 b(The)22 b(n)o(um)o(b)q(er)g(of)f(Lyndon)h(w)o(ords)f(of)h(length)g Fk(n)g Fo(is)g(\(1)p Fk(=n)p Fo(\))1333 2270 y Fn(P)1376 2314 y Fl(d)p Fh(j)p Fl(n)1435 2302 y Fk(\026)p Fo(\()p Fk(d)p Fo(\))p Fk(q)1544 2286 y Fl(n=d)1602 2302 y Fo(,)h(where)125 2359 y Fk(\026)18 b Fo(is)h(the)f(M\177)-23 b(obius)19 b(function.)29 b(\(This)19 b(w)o(ell-kno)o(wn)g(n)o(um)o(b)q(er)f(coun) o(ts)g(sev)o(eral)h(other)e(things,)125 2415 y(for)f(example,)h (irreducible)j(p)q(olynomials)e(o)o(v)o(er)e Fj(F)992 2422 y Fl(q)1029 2415 y Fo(if)h Fk(q)i Fo(is)e(a)f(prime)i(p)q(o)o(w)o (er;)e(see)h([12)o(].\))24 b(The)125 2472 y(follo)o(wing)15 b(com)o(binatorial)h(prop)q(erties)g(hold)g(for)f(Lyndon)h(w)o(ords:) 125 2588 y Fc(Lemma)h(2.2)45 b Ff(\(i\))16 b(A)o(ny)g(wor)n(d)h Fk(w)h Ff(has)f(a)g(unique)g(expr)n(ession)e(in)i(the)g(form)g Fk(w)d Fo(=)g Fk(w)1608 2595 y Fm(1)1628 2588 y Fk(w)1661 2595 y Fm(2)1687 2588 y Fk(:)8 b(:)g(:)o Ff(,)238 2645 y(wher)n(e)17 b Fk(w)400 2652 y Fm(1)419 2645 y Fk(;)8 b(w)473 2652 y Fm(2)491 2645 y Fk(;)g(:)g(:)g(:)14 b Ff(ar)n(e)i(Lyndon)f(wor)n(ds)i(with)g Fk(w)1083 2652 y Fm(1)1114 2645 y Fi(\025)c Fk(w)1195 2652 y Fm(2)1227 2645 y Fi(\025)g Fk(:)8 b(:)g(:)n Ff(.)147 2749 y(\(ii\))21 b(Given)g(Lyndon)g(wor)n(ds)h Fk(w)709 2756 y Fm(1)728 2749 y Fk(;)8 b(w)782 2756 y Fm(2)801 2749 y Fk(;)g(:)g(:)g(:)19 b Ff(with)j Fk(w)1033 2756 y Fm(1)1075 2749 y Fi(\025)h Fk(w)1166 2756 y Fm(2)1208 2749 y Fi(\025)f Fk(:)8 b(:)g(:)o Ff(,)23 b(the)f(lexic)n(o)n(gr)n(aphic)n(al)r(ly)238 2805 y(gr)n(e)n(atest)16 b(shu\017e)g(of)g(these)g(wor)n(ds)h(is)e(the) i(c)n(onc)n(atenation)e Fk(w)1290 2812 y Fm(1)1309 2805 y Fk(w)1342 2812 y Fm(2)1369 2805 y Fk(:)8 b(:)g(:)n Ff(.)p eop %%Page: 5 5 5 4 bop 140 162 a Fe(The)15 b(algebra)g(of)g(an)g(age)1159 b Fo(5)191 311 y(Hence,)17 b(if)g(w)o(e)f(let)h(\\connected")g(mean)f (\\Lyndon)h(w)o(ord",)e(\\in)o(v)o(olv)o(emen)o(t")h(mean)h(\\lex-)125 368 y(icographic)f(order)f(rev)o(ersed",)g(and)g(\\comp)q(osition")h (mean)f(\\concatenation)g(in)i(decreasing)125 424 y(lexicographic)h (order",)d(then)i(A1)f(and)g(A2)g(hold,)h(and)f(w)o(e)g(conclude)i (that)e Fi(A)p Fo(\()p Fk(A)1543 408 y Fh(\003)1563 424 y Fo(\))e(=)g Fi(A)1680 408 y Fl(G)p Fm(\()p Fl(q)q Fm(\))125 481 y Fo(is)h(a)f(p)q(olynomial)j(algebra)e(generated)g(b)o(y)g(the)g (Lyndon)g(w)o(ords)f(\(a)h(result)g(of)f(Radford)h([11]\).)125 630 y Fp(3)66 b(T)-6 b(ransitiv)n(e)25 b(extensions)125 732 y Fo(Not)19 b(m)o(uc)o(h)h(is)h(kno)o(wn)f(in)h(general)g(ab)q(out) f(ho)o(w)g(the)g(algebra)h Fi(A)1284 716 y Fl(G)1334 732 y Fo(is)f(a\013ected)g(b)o(y)h(group-)125 789 y(theoretic)f(or)f (mo)q(del-theoretic)j(constructions)e(\(direct)h(pro)q(ducts)f(with)g (pro)q(duct)h(action,)125 845 y(wreath)11 b(pro)q(ducts,)i(co)o(v)o (ers)f(and)h(quotien)o(ts,)g(etc.\).)18 b(This)13 b(section)g(con)o (tains)g(some)f(commen)o(ts)125 902 y(ab)q(out)j(transitiv)o(e)g (extensions.)191 958 y(The)g(p)q(erm)o(utation)g(group)g Fk(H)j Fo(on)d(\012)f(is)i(a)e Ff(tr)n(ansitive)h(extension)f Fo(of)h Fk(G)f Fo(if)i Fk(H)i Fo(is)d(transitiv)o(e)125 1015 y(and)20 b(the)g(stabiliser)i Fk(H)537 1022 y Fl(\013)582 1015 y Fo(of)e(the)g(p)q(oin)o(t)h Fk(\013)p Fo(,)h(acting)e(on)g(\012) 14 b Fi(n)f(f)p Fk(\013)p Fi(g)p Fo(,)21 b(is)g(isomorphic)g(to)f Fk(G)g Fo(as)125 1071 y(p)q(erm)o(utation)f(group.)31 b(Note)19 b(that,)g(in)h(this)f(situation,)i Fk(H)h Fo(is)e(closed)g (if)f(and)h(only)f(if)h Fk(G)f Fo(is)125 1127 y(closed.)191 1184 y(A)c(general)h(question:)21 b Ff(L)n(et)16 b Fk(H)k Ff(b)n(e)c(a)h(tr)n(ansitive)e(extension)h(of)g Fk(G)p Ff(.)22 b(What)17 b(is)f(the)h(r)n(elation)125 1240 y(b)n(ETWE)N(EN)E FI(A)327 1224 Y FL(H)377 1240 Y FF(和)H FI(A)501 1224πY FL(G)531 1240 Y FF(?)191 1297 y Fo(W)l(e)f(can)h(regard)f(the)g (group)g(induced)j(on)d(\012)g(b)o(y)h Fk(G)f Fo(as)g(the)g(direct)h (pro)q(duct)g(of)f Fk(G)g Fo(\(in)h(its)125 1353 y(giv)o(en)f(action\)) f(with)h(the)f(trivial)i(group)e(of)g(degree)h(1.)k(F)l(or)14 b(the)h(latter)f(group)g(\()p Fk(K)s Fo(,)g(sa)o(y\),)f(the)125 1410 y(algebra)i Fi(A)320 1393 y Fl(K)369 1410 y Fo(is)h(generated)f(b) o(y)g(an)g(elemen)o(t)h Fk(k)g Fo(of)f(degree)g(1)g(with)g Fk(k)1315 1393 y Fm(2)1348 1410 y Fo(=)e(0.)19 b(In)d(other)f(w)o (ords,)125 1466 y Fi(A)161 1450 y Fl(K)208 1454 y Fi(\030)208 1468 y Fo(=)256 1466 y Fk(T)6 b Fo(\()p Fj(Q)o Fo(\),)11 b(the)j(algebra)h(of)f(2)7 b Fi(\002)i Fo(2)14 b(upp)q(er)h(triangular) g(matrices)f(with)g(constan)o(t)g(diagonal)125 1523 y(o)o(v)o(er)g Fj(Q)p Fo(.)j(Hence,)f(using)g Fk(G)590 1506 y Fm(+)634 1523 y Fo(for)f(the)g(group)g(induced)i(on)e(\012)g(b)o(y)g Fk(G)p Fo(,)g(w)o(e)g(ha)o(v)o(e)637 1628 y Fi(A)673 1610 y Fl(G)700 1598 y Fb(+)740 1616 y Fi(\030)740 1630 y Fo(=)788 1628 y Fi(A)824 1610 y Fl(G)864 1628 y Fi(\012)899 1635 y Fd(Q)935 1628 y Fk(T)6 b Fo(\()p Fj(Q)o Fo(\))1048 1616 y Fi(\030)1048 1630 y Fo(=)1096 1628 y Fk(T)g Fo(\()p Fi(A)1183 1610 y Fl(G)1212 1628 y Fo(\))p Fk(:)125 1734 y Fo(Ho)o(w)o(ev)o(er,)18 b(w)o(e)h(can)h(only)g(sa)o(y)e(that,)h (since)i Fk(G)938 1718 y Fm(+)986 1734 y Fi(\024)f Fk(H)t Fo(,)g(the)f(algebra)g Fi(A)1397 1718 y Fl(H)1450 1734 y Fo(is)h(a)f(subalgebra)125 1791 y(of)d Fk(T)6 b Fo(\()p Fi(A)265 1774 y Fl(G)295 1791 y Fo(\).)26 b(This)18 b(do)q(es)g(not)f (seem)g(to)g(help)i(to)d(decide,)k(for)c(example,)j(whether)e Fi(A)1607 1774 y Fl(H)1659 1791 y Fo(is)g(an)125 1847 y(in)o(tegral)e(domain.)191 1904 y(There)20 b(is)h(a)f(sp)q(ecial)i (class)e(of)g(transitiv)o(e)g(extensions)h(for)e(whic)o(h)i(a)f(bit)h (more)e(can)i(b)q(e)125 1960 y(said.)h(W)l(e)16 b(sa)o(y)f(that)g(the)h (transitiv)o(e)g(extension)h Fk(H)i Fo(of)d Fk(G)f Fo(is)i Ff(curious)f Fo(if)h Fk(H)i Fo(has)d(a)f(transitiv)o(e)125 2016 y(subgroup)g(\(on)h(the)g(whole)g(of)f(\012\))h(whic)o(h)g(is)h (isomorphic)f(to)f Fk(G)p Fo(.)22 b(In)16 b(the)g(case)g(where)g Fk(G)f Fo(and)125 2073 y Fk(H)23 b Fo(are)d(closed,)h(this)g(means)f (that)f Fk(H)k Fo(is)e(a)f(reduct)g(of)g Fk(G)p Fo(.)34 b(If)20 b Fk(H)j Fo(is)e(a)e(curious)i(transitiv)o(e)125 2129 y(extension)e(of)e Fk(G)p Fo(,)i(then)f Fi(A)593 2113 y Fl(H)645 2129 y Fo(is)h(a)f(subalgebra)h(of)f Fi(A)1057 2113 y Fl(G)1086 2129 y Fo(;)i(in)f(particular,)g Fi(A)1437 2113 y Fl(H)1489 2129 y Fo(is)g(an)f(in)o(tegral)125 2186 y(domain)c(if)g Fi(A)361 2169 y Fl(G)404 2186 y Fo(is.)20 b(P)o(erhaps)14 b(it)g(is)g(p)q(ossible)i(to)d(w)o(ea)o(v)o (e)g(together)g(the)g(em)o(b)q(eddings)j(of)d Fi(A)1669 2169 y Fl(H)1717 2186 y Fo(in)125 2242 y Fi(A)161 2226 y Fl(G)206 2242 y Fo(and)i(in)h Fk(T)6 b Fo(\()p Fi(A)434 2226 y Fl(G)463 2242 y Fo(\))15 b(to)g(get)g(b)q(etter)g(information.) 125 2353 y Ff(Example)f(1)h(\(c)n(ontinue)n(d\).)j Fo(A)c Ff(two-gr)n(aph)g Fo(on)g(\012)f(is)h(a)f(set)h Fk(T)19 b Fo(of)13 b(3-elemen)o(t)h(subsets)g(of)f(\012)g(suc)o(h)125 2410 y(that)h(an)o(y)h(4-subset)g(con)o(tains)g(an)h(ev)o(en)f(n)o(um)o (b)q(er)h(of)e(mem)o(b)q(ers)i(of)e Fk(T)21 b Fo(\(Seidel)c([14]\).)191 2466 y(Giv)o(en)e(a)g(graph)g(\000)h(on)f(\012,)f(let)i Fk(T)6 b Fo(\(\000\))14 b(b)q(e)i(the)f(set)g(of)g Ff(o)n(dd)h(triples) f Fo(of)g(\000)g(\(those)g(con)o(taining)125 2523 y(an)f(o)q(dd)h(n)o (um)o(b)q(er)g(of)f(edges\).)19 b(Then)c Fk(T)6 b Fo(\(\000\))14 b(is)h(a)f(t)o(w)o(o-graph)f(on)i(\012.)k(Ev)o(ery)14 b(t)o(w)o(o-graph)f(arises)125 2579 y(in)j(this)f(w)o(a)o(y)l(.)191 2636 y(Let)j Fk(R)f Fo(b)q(e)h(the)g(random)f(graph)g(on)h(\012)871 2643 y Fm(0)890 2636 y Fo(.)27 b(T)l(ak)o(e)17 b(a)g(new)h(p)q(oin)o(t) g Fi(1)p Fo(,)g(and)g(de\014ne)h Fk(T)k Fo(to)17 b(b)q(e)125 2692 y(the)g(t)o(w)o(o-graph)f(on)h(\012)f(=)g(\012)620 2699 y Fm(0)651 2692 y Fi([)c(f1g)17 b Fo(deriv)o(ed)h(from)f Fk(R)g Fo(\(with)g Fi(1)h Fo(as)e(an)i(isolated)g(v)o(ertex\).)125 2749 y(Then)f(Aut\()p Fk(T)6 b Fo(\))16 b(is)h(a)g(transitiv)o(e)g (extension)h(of)e(Aut\()p Fk(R)p Fo(\).)24 b(Moreo)o(v)o(er,)16 b(it)h(is)g(curious;)h(for)e(the)125 2805 y(t)o(w)o(o-graph)e(deriv)o (ed)i(from)f Fk(R)g Fo(without)g(an)h(isolated)g(v)o(ertex)f(is)h (clearly)g(a)g(reduct)f(of)g Fk(R)p Fo(,)g(and)p eop %%Page: 6 6 6 5 bop 125 162 a Fo(6)1321 b Fe(P)l(.)15 b(J.)g(Cameron)125 311 y Fo(is)21 b(isomorphic)g(to)f Fk(T)6 b Fo(.)35 b(\(In)21 b(fact,)g Fk(T)26 b Fo(is)21 b(the)g(unique)h(coun)o(table)f(univ)o (ersal)h(homogeneous)125 368 y(t)o(w)o(o-graph.\))c(See)d(Thomas)g([16) o(].)125 424 y Ff(Pr)n(oblem.)k Fo(Is)d Fi(A)406 408 y Fm(Aut\()p Fl(T)5 b Fm(\))536 424 y Fo(a)15 b(p)q(olynomial)i (algebra?)125 481 y Ff(R)n(emark.)j Fo(Mallo)o(ws)14 b(and)h(Sloane)g([9])f(sho)o(w)o(ed)g(that)g(the)g(n)o(um)o(b)q(ers)h (of)f(t)o(w)o(o-graphs)f(and)i Ff(even)125 537 y(gr)n(aphs)g Fo(\(graphs)g(with)h(all)g(v)m(alencies)i(ev)o(en\))e(on)f Fk(n)h Fo(p)q(oin)o(ts)g(are)f(equal.)22 b(Hence,)16 b(if)g Fi(A)1593 521 y Fm(Aut\()p Fl(T)5 b Fm(\))1724 537 y Fo(is)125 594 y(a)18 b(p)q(olynomial)i(algebra,)f(then)g(its)f (generators)g(are)g(in)h(one-to-one)g(corresp)q(ondence)g(\(pre-)125 650 y(serving)c(degree\))g(with)h(the)f(\014nite)h Ff(Eulerian)g(gr)n (aphs)g Fo(\(the)e(connected)j(ev)o(en)e(graphs\).)125 757 y Ff(Example)20 b(3)i(\(c)n(ontinue)n(d\).)33 b Fo(Let)21 b Fk(G)p Fo(\()p Fk(q)r Fo(\))e(b)q(e)i(as)e(in)j(Example)e(3)g(in)h (the)g(preceding)g(section.)125 814 y(Then)15 b Fk(G)p Fo(\()p Fk(q)r Fo(\))f(has)h(a)g(transitiv)o(e)h(extension)g Fk(H)t Fo(\()p Fk(q)r Fo(\))e(de\014ned)i(as)f(follo)o(ws.)191 870 y(On)g(the)f(set)f(of)h(complex)h(ro)q(ots)e(of)h(unit)o(y)l(,)g (put)g Fk(z)1034 877 y Fm(1)1067 870 y Fi(\021)f Fk(z)1136 877 y Fm(2)1170 870 y Fo(if)h Fk(z)1231 877 y Fm(2)1251 870 y Fk(z)1274 851 y Fh(\000)p Fm(1)1272 883 y(1)1335 870 y Fo(is)h(a)e Fk(q)r Fo(th)h(ro)q(ot)f(of)h(unit)o(y)l(.)125 927 y(Let)e(\012)h(b)q(e)g(a)g(dense)g(subset)g(con)o(taining)h (exactly)f(one)f(mem)o(b)q(er)h(of)g(eac)o(h)f(equiv)m(alence)k(class)d (of)125 983 y(this)j(relation.)21 b(\(Suc)o(h)16 b(a)g(set)f(is)h 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Fo(a)g(p)q(oin)o(t)h(on)f (the)h(unit)g(circle)h(whic)o(h)f(is)g(not)f(a)g(ro)q(ot)g(of)g(unit)o (y)l(,)h(w)o(e)f(obtain)h(a)f(bijection)i(from)125 1902 y(all)e(of)e(\012)h(to)g(a)g(coun)o(table)h(dense)g(subset)f(of)g(\(0)p Fk(;)8 b Fo(1\))13 b(partitioned)j(in)o(to)f Fk(q)i Fo(dense)f (subsets.)125 1958 y Ff(Pr)n(oblem.)j Fo(Is)d Fi(A)406 1942 y Fl(H)s Fm(\()p Fl(q)q Fm(\))499 1958 y Fo(a)f(p)q(olynomial)i (algebra?)125 2015 y Ff(R)n(emark.)h Fo(F)l(or)10 b Fk(q)k Fo(=)f(2,)e(the)f(relations)h Fk(R)803 2022 y Fm(1)832 2015 y Fo(and)f Fk(R)950 2022 y Fm(2)980 2015 y Fo(are)g(a)f(con)o(v)o (erse)h(pair)h(of)e(tournamen)o(ts,)h(eac)o(h)125 2071 y(of)g(whic)o(h)h(is)g(isomorphic)g(to)f(the)g(coun)o(table)h(univ)o (ersal)h(homogeneous)e Ff(lo)n(c)n(al)h(or)n(der)g Fo([2)o(],)f Ff(lo)n(c)n(al)r(ly)125 2128 y(tr)n(ansitive)j(tournament)h Fo([7)o(],)g(or)e Ff(vortex-fr)n(e)n(e)j(tournament)f Fo([6)o(]:)19 b(these)14 b(are)f(three)g(alternativ)o(e)125 2184 y(names)i(for)g(a)g(tournamen)o(t)g(ha)o(ving)g(no)h(subtournamen) o(t)f(consisting)h(of)f(a)h(directed)g(3-cycle)125 2240 y(dominating)i(or)g(dominated)h(b)o(y)f(a)g(v)o(ertex.)28 b(This)19 b(structure)f(is)h(further)f(discussed)h(in)g(the)125 2297 y(lectures)d(of)e(Ev)m(ans,)h(Iv)m(ano)o(v)h(and)f(Macpherson.)191 2353 y(Orbits)20 b(of)f Fk(H)t Fo(\()p Fk(q)r Fo(\))f(on)h Fk(n)p Fo(-sets)g(are)g(parametrised)h(b)o(y)f(t)o(w)o(o-w)o(a)o(y)e (in\014nite)k(\\shift)e(register)125 2410 y(sequences")d(\()p Fk(x)398 2417 y Fl(i)412 2410 y Fo(\))f(with)h(elemen)o(ts)g(in)g Fi(f)p Fo(1)p Fk(;)8 b(:)g(:)g(:)d(;)j(q)r Fi(g)14 b Fo(satisfying)i Fk(x)1224 2417 y Fl(i)1248 2410 y Fo(+)11 b Fk(n)j Fi(\021)f Fk(x)1409 2417 y Fl(i)1434 2410 y Fo(+)d(1)23 b(\(mo)q(d)15 b Fk(q)r Fo(\))g(for)125 2466 y(all)21 b Fk(i)p Fo(.)35 b(F)l(or)20 b Fk(q)j Fo(=)e(2,)g(the)g (sequences)g(coun)o(ting)g(these)f(orbits)h(is)g(listed)g(as)f(M0324)f (in)i(the)125 2523 y Ff(Encyclop)n(e)n(dia)15 b(of)h(Inte)n(ger)f(Se)n 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b(not)f(homogeneous;)i(indeed,)i(the)d(class)g(of) f(CC-structures)h(\(or)f(of)g(represen)o(table)i(CC-)125 819 y(structures\))14 b(do)q(es)i(not)e(ha)o(v)o(e)h(the)h (amalgamation)e(prop)q(ert)o(y)l(.)191 876 y(Giv)o(en)21 b(a)f(ternary)f(relation)i Fk(R)f Fo(on)g(\012)g(whose)g(restriction)h (to)f(an)o(y)g(3-set)g(is)g(a)g(circular)125 932 y(order,)e(there)h(is) g(a)g(deriv)o(ed)g(tournamen)o(t)f Fk(R)918 939 y Fl(\013)961 932 y Fo(on)h(\012)12 b Fi(n)g(f)p Fk(\013)p Fi(g)18 b Fo(de\014ned)i(b)o(y)f Fk(R)1463 939 y Fl(\013)1487 932 y Fk(\014)r(\015)i Fi(,)e Fk(R\013\014)r(\015)s Fo(.)125 989 y(Kn)o(uth's)12 b(\014fth)h(axiom)g(for)f(CC-structures)g(implies)j (that)d Fk(R)1164 996 y Fl(\013)1201 989 y Fo(is)h(a)f(lo)q(cal)i (order)e(for)h(an)o(y)f(p)q(oin)o(t)125 1045 y Fk(\013)p Fo(.)21 b(Indeed,)c(if)f(w)o(e)g(tak)o(e)f(the)h(univ)o(ersal)g (represen)o(table)h(CC-structure)e(ab)q(o)o(v)o(e,)g(and)h(pro)s(ject) 125 1102 y(\012)c Fi(n)h(f)p Fk(\013)p Fi(g)19 b Fo(radially)i(on)o(to) 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y(Math.)i(So)n(c.)f Fo(\(2\))f Fc(23)i Fo(\(1981\),)d(249{265.)147 1961 y([3])22 b(P)l(.)10 b(J.)h(Cameron,)f Ff(Oligomorphic)j(Permutation)f(Gr)n(oups)p Fo(,)g(London)f(Math.)e(So)q(c.)i(Lecture)218 2017 y(Notes)k Fc(152)p Fo(,)g(Cam)o(bridge)g(Univ)o(ersit)o(y)h(Press,)f(Cam)o (bridge,)f(1990.)147 2107 y([4])22 b(P)l(.)16 b(J.)g(Cameron,)f(The)h (random)g(graph,)g(pp.)g(333{351)e(in)j Ff(The)g(Mathematics)g(of)g (Paul)218 2164 y(Er)n(d})-23 b(os,)17 b(II)e Fo(\(ed.)g(R.)h(L.)g (Graham)f(and)h(J.)g(Ne)n(\024)-20 b(set)n(\024)g(ril\),)14 b(Algorithms)i(and)g(Com)o(binatorics)218 2220 y Fc(14)p Fo(,)f(Springer,)h(Berlin,)g(1997.)147 2310 y([5])22 b(J.)e(Co)o(vington,)h(A)f(univ)o(ersal)h(structure)f(for)g(N-free)g (graphs,)h Ff(Pr)n(o)n(c.)f(L)n(ondon)g(Math.)218 2366 y(So)n(c.)14 b Fo(\(3\),)g Fc(58)i Fo(\(1989\),)d(1{16.)147 2456 y([6])22 b(D.)e(E.)h(Kn)o(uth,)i Ff(Axioms)e(and)h(Hul)r(ls)p Fo(,)f(Lecture)h(Notes)f(in)h(Computer)e(Science)j Fc(606)p Fo(,)218 2513 y(Springer,)16 b(Berlin,)g(1992.)147 2602 y([7])22 b(A.)g(H.)f(Lac)o(hlan,)j(Coun)o(table)e(homogeneous)g (tournamen)o(ts,)g Ff(T)m(r)n(ans.)f(A)o(mer.)h(Math.)218 2659 y(So)n(c.)14 b Fc(284)p Fo(,)i(431{461.)147 2749 y([8])22 b(A.)16 b(H.)h(Lac)o(hlan)h(and)f(R.)g(E.)f(W)l(o)q(o)q(dro)o (w,)g(Coun)o(table)h(ultrahomogeneous)g(undirected)218 2805 y(graphs,)d Ff(T)m(r)n(ans.)h(A)o(mer.)h(Math.)g(So)n(c.)f Fc(262)h Fo(\(1980\),)d(51{94.)p eop %%Page: 8 8 8 7 bop 125 162 a Fo(8)1321 b Fe(P)l(.)15 b(J.)g(Cameron)147 311 y Fo([9])22 b(C.)15 b(L.)h(Mallo)o(ws)g(and)g(N.)f(J.)h(A.)f (Sloane,)i(Tw)o(o-graphs,)d(switc)o(hing)j(classes,)f(and)g(Euler)218 368 y(graphs)f(are)g(equal)h(in)g(n)o(um)o(b)q(er,)f Ff(SIAM)g(J.)h(Appl.)g(Math.)f Fc(28)h Fo(\(1975\),)d(876{880.)125 462 y([10])21 b(P)l(.)14 b(M.)h(Neumann,)g(The)g(la)o(wlessness)g(of)g (\014nitary)g(p)q(erm)o(utation)g(groups,)f Ff(A)o(r)n(ch.)h(Math.)218 518 y Fc(26)g Fo(\(1975\),)e(561{566.)125 612 y([11])21 b(D.)14 b(E.)g(Radford,)h(A)g(natural)f(ring)h(basis)h(for)e(the)g(sh)o (u\017e)h(algebra)g(and)g(an)g(application)218 668 y(to)g(group)f(sc)o (hemes,)i Ff(J.)g(A)o(lgebr)n(a)e Fc(58)h Fo(\(1979\),)e(432{454.)125 762 y([12])21 b(C.)j(Reutenauer,)j Ff(F)m(r)n(e)n(e)d(Lie)g(A)o(lgebr)n (as)p Fo(,)h(London)g(Math.)f(So)q(c.)g(Monographs)g(\(New)218 819 y(Series\))16 b Fc(7)p Fo(,)f(Oxford)g(Univ)o(ersit)o(y)h(Press,)e (1993.)125 912 y([13])21 b(G.-C.)14 b(Rota,)g(On)i(the)f(foundations)g (of)g(com)o(binatorial)g(theory)l(,)g(I:)g(Theory)f(of)h(M\177)-23 b(obius)218 969 y(functions,)16 b Ff(Z.)g(Wahrscheinlichkeitsthe)n (orie)e Fc(2)i Fo(\(1964\),)d(340{368.)125 1063 y([14])21 b(J.)16 b(J.)g(Seidel,)h(A)f(surv)o(ey)g(of)g(t)o(w)o(o-graphs,)e(pp.)i (481{511)e(in)i Ff(Pr)n(o)n(c.)h(Int.)f(Col)r(lo)n(q.)g(T)m(e)n(orie) 218 1119 y(Combinatorie)p Fo(,)f(Accad.)g(Naz.)g(Lincei,)i(Roma,)d (1977.)125 1213 y([15])21 b(N.)c(J.)g(A.)g(Sloane)h(and)f(S.)g (Plou\013e,)h Ff(The)g(Encyclop)n(e)n(dia)f(of)h(Inte)n(ger)f(Se)n (quenc)n(es)p Fo(,)e(Aca-)218 1269 y(demic)h(Press,)f(New)g(Y)l(ork,)g (1995.)125 1363 y([16])21 b(S.)f(R.)h(Thomas,)g(Reducts)g(of)g(the)f (random)g(graph,)i Ff(J.)f(Symb)n(olic)f(L)n(o)n(gic)f Fc(56)i Fo(\(1991\))218 1420 y(176{181.)125 1526 y Fc(Author's)16 b(address:)125 1582 y Fo(Sc)o(ho)q(ol)f(of)g(Mathematical)g(Sciences,) 125 1639 y(Queen)h(Mary)e(and)i(W)l(est\014eld)g(College,)125 1695 y(London)f(E1)g(4NS,)125 1752 y(England.)125 1808 y Fc(e-mail:)44 b Fo(P)l(.J.Cameron@qm)o(w.ac.uk)p eop %%Trailer e如果%%EOF,则ND用户标识/结束钩子已知{结束钩子}