Sonderabdru<k au.

ARCHIV DER MATHEMATIK

Vol. XIX, 1968 BIRKHAUSER VERLAG, BASEL UND STUTTGART Fase. 6 -------------.. -----.--- --- .---.------.--.-... -. -----------

Mobius Inversion in Lattices

By

HENRY H. CRAl'Ol)

1. Introduction. In the development of computational techniques for combinatorial theory, attention has lately centered on ROTA'S theory of Mobius inversion [6]. The main theorem of ROTA'S paper, concerning the computation of the Mobius invariant across a Galois connection, is a prerequisite to the use of lattice-theoretic methods in combinatorics.

By suitably combining ROTA'S main theorem with a discrete analogue of inte­gration-by-parts, we here obtain a perfectly general formulation of Mobius inversion across a Galois connection (theorem 3, below).

As immediate applications of this theory, we obtain a number of interesting computational results concerning finite lattices (section 3, 4) and combinatorial geometries (section 5).

2. Mobius Inversion across a Galois Connection. We begin with a restatement and a simplified proof of ROTA'S main theorem. The proof tlli"lls on the essential fact that for any (locally finite) ordered set Q with least element 0, the recursion

.2;a(y) "y, z) = 0 for H 0 I/EQ

has the unique solution a(y) = 0 with initial condition a(O) = 0, and has the unique solution a (y) = #Q (0, y) with initial condition a (0) = 1. Recall that the zeta function "y, z) has value 1 if Y ~ z, and has value 0 otherwise.

Theorem 1. If J is a closure operator on a finite lattice P, and Q = P/J is the quotient lattice, consisting of the J-closed elements of P, then for all elements x E P, and elements y closed in P, x ~ y, the sum

L #(x, t) t;z&;t&;J(t)-1I

has value #Q (x, y) if X is closed, and has value 0 otherwise.

1) We wish to express our gratitude to the National Research Council, Canada, for their support of this research (grant A-2994), to K. JACOBS, for his organization of the extraordinary conference "Kombinatorik" at Oberwolfach, and to D. KLEITMAN and J. GOLDMAN, for their organization of the combinatorics seminar at M.I.T., for which this material was prepared.

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596 H. H. CRAPO ARCH. MATH.

Proof. Note that the theorem may be rewritten in the form

(1) 15 (X, J(X)) PQ(J(X), y) = LP(X, I) 15 (J(/), y). tE?

Without loss of generality, we assume x = 0 in P. For each element y E Q, let a(y) = L p(O, I) 15 (J(/), y). Then

tEP

L a(y) C(y, z) = L p(O, I) b(J (I), y) C(y, z) = L p(O, I) C(/, z) = bp(O, z) . IIEQ t.1I EP t EP

IfO<J(O), bp(O,z) =0 for all ZEQ, and a(y)=O for all YEQ. If O=J(O), bp(O, z) = 1 for Z = 0, and a(y) = PQ(O, y). I

Given a function f from a finite lattice P into a ring with unit, associate the difference operators D, E lower difference Df(x) = Lf(y)p(y, x),

lI;II""X

upper difference Ef(x) = LP(x, y)f(y)· 1I;:Z:~Y

Theorem 2 (Analogue of integration by parts). If f, g are funclions from a finile lattice P inlo a ring, then

LDf(x) g(x) = Lf(x) Eg(x). xeP xeP

Proof. Both are equal to Lf(x)p(x,y)g(y). I X.II

It is interesting to compare the proof of theorem 2 with the argument that cycles and coboundaries in a graph are orthogonal to one another. For each vertex p and edge x, let

1+ 1 if p is the head of x,

f (p, x) = - 1 if P is the tail of x, o otherwise.

Boundary and co boundary operators are defined by

af(p) = L f(p, x) f(x) for any 1-chain f, x

bg(x) = Lg(P) f(p, x) for any O-chain g. p

If f is a 1-cycle (of = 0) and h is a 1-coboundary (h = bg), then

Lf(x)h(x) = Lf(x)f(p,x)g(p) = Laf(p)g(p) = LOg(P)=O. x x,p p p

If a: P -+ L is a supremum-homomorphism from a complete lattice P into a complete lattice L, then ad: L -+ P is an infimum-homomorphism, defined by

ad(y) = sup{x; a (x) ~ y}.

The pair a, ad is a Galois connection, in the sense that P/ad (a) is isomorphic to L/a(ad), where ad (a) is a closure operator on P and a(ad) is a co closure operator on L. All Galois connections between complete lattices arise in this fashion. In the special case where a is onto L, P/ad (a) ~ L.

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Vol. XIX. 1968 Mobius Inversion in Lattices 597

We now combine theorems 1 and 2 to establish a general theorem on Mobius inversion across Galois connections. This theorem is the discrete analogue of the change-of-variables formula for integration. and of Stokes' Theorem, f w = f dw.

as s

Theorem 3. If 0': P --+ L i8 a 8up-homomorphism from a finite lattice P into a finite lattice L, if f i8 a function from P and g i8 a function from L taking values in a ring, then

'2,Df(x)g(O'(x» = '2, f(O'-1 (y» Eg(y) . ZEP 1/EL

Proof. Let Q = P/O'-1O' ~ L/O'O'-1 be the common quotient lattice, and regard both f and g, restricted to closed elements in P and L, as functions on Q. Then

'2, Df(x) g (0' (x» = '2, f(t) f-l(t, x) g (0' (x)) = '2, f(t)f-lp(t,x)b(0'(X),8)g(8) = ZEP t,ZEP t,zeP,seQ

(2) = '2, f(r) f-lQ(r, 8) g(8) = '2, f(r)b(r,O'-1(Y»f-ldy,z)g(z)= r,seQ reQ,',1/EL

= '2, f(O'-1(y))f-lL(y,z)g(z) = '2,f(O'-1(y))Eg(y). I 1/,'EL 1/EL

A number of related forms of Theorem 3 may be more convenient in applications. For instance, using the fact that the difference operators D and E are inverses to the summation operators Sand T,

Sf(x) = '2, f(y); Tf(x) = '2, f(y)' 1/;1/;;>Z 1I;a;~11

we obtain corollary 1 by substituting Sf for /. Tg for g.

Corollary 1. ItO': P --+ L i8 a sup-homomorphism from a finite lattice P into a finite lattice L, if f is a function on P and g is a function on L, then

'2,f(x) g(O'(x)) = '2, Sf (0'-1 (y)) Eg(y), ZEP 1/EL

'2, Df(x) Tg(O'(x)) = '2,f(O'-1(y))g(y), ZEP IIEL

'2,f(x) Tg(O'(x)) = '2,Sf(O'-1(y))g(y). I ZEP 1/EL

The symmetric intermediate form (2) appearing in the proof of Theorem 3 deserves special note:

Corollary 2. If 0': P --+ L is a sup-homomorphism from a finite lattice P into a finite lattice L, if Q is the quotient lattice P/O'-1 (0') ~ L/O'(~), and if functions f on P and g on L are defined on Q by restriction to closed elements of P, coclosed elements of L, respectively, then

'2,Df(x)g(O'(x» = '2,f(r) f-lQ(r, s)g(s) = '2, f(O'-1 (y)) Eg(y) . I XEP r,seQ 1/EL

For Galois connections given directly as a pair of order-inverting maps 0', T whose composites O'T and TO' are increasing, it is more convenient to have Theorem 3 in the form obtained by inverting the lattice L, as follows.

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598 H. H.CRAPO ARCH. MATH.

Corollary 3. If rJ: P -+ L, i: L -+ P is a Galois connection between finite lattices P and L, if f is a function on P, and g is a function on L, then

LDf(x)g(rJ(x)) = Lf(i(y))Dg(y).1 ",eP lIeL

Theorem 1, above, lacks the full symmetry of Galois connections because it operates between a lattice P and its quotient Q, rather than between two lattices P, L, with a common quotient Q. The symmetric form for Theorem 1, and thus for ROTA'S main theorems, is recoverable from Theorem 3 as follows.

Corollary 4. If a is a sup-homomorphism from a finite lattice P into a finite lattice L, then for any elements t E P, Z E L

Lit (t, x) b (a(x), z) = L b (t, aLi (y)) It (y, z) . "'EP veL

This common value is clearly equal to 0 unless t is closed in P (ie: t < x => a (t) < a (x)) and Z is coclosed in L (ie: 3x E P; a(x) = z). If t is closed in P and z is closed in L, both t and z correspond to elements of the common quotient lattice Q, and the common value of the summations is equal to ItQ(t, z).

Proof. Setf(x) = b(t,x), g(y) = b(y,z). Then Df(x) = ItP(t,x) and Eg(y) = ltL(y,z). The intermediate symmetric form (2) arising in the proof of Theorem 3 is in this case equal to

L b(t, r) ItQ(r, s) b(s, z). I " .. Q

The theory of Mobius inversion across a composite of sup-homomorphisms develops directly from Theorem 3.

Corollary 5. If a,: P'-l -+ PI, i = 1, ... , k, is a sequence of sup-homomorphisms between finite lattices, if f is a function on Po and g is a function on Pk, then the sums

(3) L f (at ( ... af (x) )) It (x, y) g (ak ( ... at+l (y) )) f£,lIEPi

are equal, fori = 0, ... , k. (Note thatfori = k the evaluation of g is at y.) I For computations involving composites such as aLi (ill) it should be borne in mind

that Lf is a contravariant functor, ie: aLi (i.1) = (i(a))LI. The applicability of Corollary 5 is appreciably extended by the observation that

composites of sup-homomorphisms give rise to commutative diagrams involving the intermediate quotients. For two sup-homomorphisms, diagram 1 applies.

Diagram 1

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Vol. XIX, 1968 Mobius Inversion in Lattices 599

For any sup-homomorphism p, the symbol PI indicates the map of each element to its closure, regarded as an element of the quotient lattice, while the symbol P2 indicates the map of each element of the quotient, regarded as a closed element of the domain, to its image under p. Note that

(a Th = al(a2 Tlh and (a T)2 = (a2 Tllz T2.

A result not obvious from previous forms of Theorem 3 derives from such consid· eration of quotients. Note that the lattices Land Q in the following corollary are not related by a Galois connection.

Corollary 6. If a: P -+ L and T: L -+ Mare sup·homorrwrphisms between finite lattices, if f is a function on P and g is a function on M, and if Q is the common quotient lattice 0/ P and M relative to the composite -r(a), then

L f(a4 (x))PL(x,y)g(T(Y)) = Lf(r)PQ(r,s)g(s). I $.lIeL r.8eQ

The expressions f(r), g(s) in Corollary 6 refer as usual to f((T(a))f (T)) and g((T(a))2 (s)), the values of f and g at elements closed in L, coclosed in M, relative to T(a).

3. Enumerative Lattice Theory. To each binary relation !?: X -+ Y between finite sets X and Y there corresponds a Galois connection (a, T) between the Boolean algebras B(X), B(Y). For all A ~ X, B ~ Y,

a(A) = {YE Y; xEA => X!?Y},

T(B)={XEX; YEB=>x!?y}.

(These definitions are simply C1(A) = nA, C1(B) = nB, if elements of X are viewed as subsets of Y, and elements of Yare viewed as subsets of X.)

Theorem 4. If !?: X -+ Y is a relation between finite sets, if f is a function defined on subsets of X, and if g is a function defined on subsets of Y, then

LDf(A)g(nA ) = Lf(nB)Dg(B). A'X B~Y

Proof. Apply Corollary 3 of Theorem 3 to the Galois connection B(X) ~ B(Y} defined by C1(A} = nA ~ Y for A ~ X, T(B) = nB~X for B~ Y. I •

The elements of the common quotient lattice Q are precisely those pairs (A, B) A ~X, B~ Y, which are

1} totally related: xEA, yE B => X!?y and maximal, in the sense that 2} x ¢ A=>3 Y E B, x e y , 3} y¢ B => 3XEA, xey, where e denotes negation of relation !?

Each element SEQ thus has a cardinality Isl1 as a subset of X and a cardinality Isl2 as a subset of Y.

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600 H. H. CRAPO ARCH. MATH.

Corollary 1. If (a, -r) is the Galois connection defined by a finite relation e: X -4- Y,

2: (rp - 1)IAI vla(A)1 = 2: rpl«B)1 (v - 1)IBI. A~X B~Y

This sum may in turn be calculated on the common quotient lattice Q, and is equal to

2: rpl'll P,Q (r, s) vl•I •• ',SEQ

Proof. Let f(A) = rplAI for all A ~X, and g(B) = vlBI , for all B~ Y. Noting that Df(A) = 2: rp IOI(- 1)IA-o l = (rp - 1)IAI, and similarly for Dg, the result

O~A

follows directly from Theorem 4. I

Modulo a few redundancies, Corollary 1 to Theorem 4 is also the fundamental enumerative structure theorem for finite lattices. The redundancies arise, causing nonisomorphic relations to have isomorphic lattices, when a(x) = a(A), for some subset A ~ X and some element x ¢; A (also when this situation occurs for some element and subset of Y). When such redundancies do not occur in the relation e, the supremum-irreducible elements of the lattice Q are precisely the pairs (x, a(x)) for x E X, the infimum-irreducible elements of Q are precisely the pairs (-r(y), y) for y E Y, and the relation e may be recovered from the lattice Q by

(4) xey~x? yin Q.

Corollary 2. Let Q be a finite lattice, with set X of supremum-irreducible elements and set Yof infimum-irreducible elements. For each element Z EQ, let IX (z), (3(z) be the mtmbers of sup-irreducibles beneath z and inf-irreducibles above z, respectively. Then

2: (rp - 1)IAI PP(supA) = 2: rpcc(lnfB) (v - 1)1 BI, A~X B~Y

and both sums are equal to 2:rpcc(')p,(r,s)PP(·). I

.,seQ

Redundancies can be reintroduced on the other side of the relation -- lattice cor­respondence, with interesting results. Given a finite lattice L, and functions j and k from finite sets X and Y, respectively, into L, a binary relation e is defined by

(5) xey~j(x) ?k(y).

Let Q be the quotient lattice of the Galois connection determined by the relation e. Then Q has two order-embeddings in L, neither of which is associated with a closure operator on L. Corollary 6 to Theorem 3 applies.

Theorem 5. Let j and k be functions from finite sets X and Y into a finite lattice L, and let e and Q be the relation and quotient lattice described above. Each element z E L has cardinalities I z 11 and I z 12 given by

Izll = l{eEX; j(e)? z}l, Izl2 = l{eEY; z? k(e)}l.

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Vol. XIX, 1968 Mobius Inversion in Lattices 601

Each element SEQ is realized as a pair (A, B) of subsets of X, Y, and thus has car­dinalities Is II = I A I, Isl2 = I B I·

L IP lx " ,uL(x, y) Villi. = L IPIT" ,uQ(r, s) vlsl •. x,yeL r,seQ

In particular, ,uQ(O, 1) = L ~(O, Ixll),u(x,y)~(jYI2'0). x,veL

Proof. The function j extends to a sup-homomorphism a from the Boolean algebra B(X) into L by alA) = supL{j(e); eEA}. Similarly, k extends to an inf­homomorphism .LI from the inverted Boolean algebra S (Y) into L (with opposite

.: L-+ SlY»~, defined by .LI(B) = infL{k(e); eEA}. Then

.(a(A» = {bEY; aEA *j(a) ~ k(b)},

and the quotient lattice with respect to the composite .(a) is equal to Q. The formula follows from Corollary 6 to Theorem 3, and the special case results from setting 11' = Y = 0, realizing that I r 11 = 0 * r = 0 E Q and Isl2 = 0 * s = 1 E Q . I

If, in the situation described above, X = Y = L, and if L is assumed to be an ordered set, not necessarily a lattice, then the resulting lattice Q is the MACNEILLE completion of the ordered set L.

4. Cross-cuts and Complementation. ROTA'S cross-cut theorem [6] and this author's complementation theorem [1] have in common a double application of Mobius in­version. Interesting sidelights on these theorems are obtainable by consideration of a lattice of the intervals of a finite lattice.

Theorem 6. Given a finite lattice L, let I (L) be the set consisting of the empty inter­val 0, together with all intervals [x, y], for x ~ y in L, ordered by containment. Then f'I(L) ([x, y], [w, z]) = ,uL(W, x) ,uL(y, z) if w ~ x ~ y ~ z, and

,uI(L) (0, [x, y]) = - ,uL(x, y).

Proof. If w ~ x ~ y ~ z, the interval from [x, y] to [w, z] is isomorphic to the

cartesian product of the inverted interval [w, xl in L with the interval [y, z] in L. But ,uL(Y), x) = ,uL(X, w), and the Mobius invariant is multiplicative on cartesian products. This establishes the product formula.

,uI(L) (0, [w, z]) = - L ,uI(L) ([x, y], [w, z]) = - L ,uL(W, x) ,uL(y, z) = :t,y;w;:£;x:;;;;;y;:;;;z x.Y;W~X~1I~Z

= - L ,uL(w, x) ~ (x, z) = - ,uL(W, z). I x;w~z;;;;;z

Theorem 7. Given a finite lattice L, an arbitrary subset X ~ L, a function f defined on subsets of X, and a function g defined on intervals of L, then

LDf(A)g([inf A, sup A]) = A~X

= f(0) g(0) - f(0) L ,u(w, z) g([w, z]) + L fIX () [x, y]) ,u(w, x) C(x, y) ,u(y, z) g([w, z]). w,zeL W,x.'V,zeL

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602 H. H. CRAPO ARCH. MATH.

Proof. The map a defined by a(A) = [inf A, sup A] is a sup-homomorphism from the Boolean algebra B (X) into the interval lattice I (L), because

a(A u B) = [inf(A u B), sup (A u B)] = [inf A, sup A] v [inf B, sup B].

Note that ad ([x, y]) = X n [x, y]. By Theorem 3,

LDf(A)g([infA, sup A]) = A,X

L f(X n [x, y]) P,I(L) ([x, y], [w, z])g([w, z]), o ;;;; ["'. v] ;;;;[w.z] ;;;;[0.1]

which reduces to the required form, by Theorem 6, once the summation is separated into three parts:

0=[x,y]=[w,z], 0=[x,y]<[w,z], and 0<[x,y]~[w,z]. I Corollary 1. If X and Yare arbitrary subsets of a finite lattice L, let qk be the number

of k-element subsets A of X disjoint from Yand spanning L (ie: inf A = 0, sup A = 1). Then

qo - ql + qz - ... =

= !5L (O, 1) - p,dO, 1) + 2: C(X n [x, y], Y) p,(0, x)C(x, y)p,(y, 1). "'.VEL

Proof. Set f(A) = C(A, Y), so that

Df(A) = 2:(-l)IAI-IBIC(B, Y) = (-1)IAI!5(0,An Y). B,A

Set g([w, z]) = 15(0, w) !5(z, 1). The sinister of the equation in Theorem 7 becomes 00

2: (-1)IAI!5(0, A n Y)!5(O, infA) 15 (sup A, 1) = 2: (-l)kqk, A,X k=O

and the simplification of the dexter is obvious. I The cross-cut theorem, the complementation theorem, and, one may conjecture,

other interesting facts about Mobius invariants of lattices are evaluations of Corol­lary 1 at particular sets X, Y.

Corollary 2 (The Cross-cut Theorem). If X is a cross-cut of a finite lattice L, and if qk is the number of k-element subsets of X which span L, then

qo - ql + qz - ... = p,dO, 1).

Proof. In Corollary 1 to Theorem 7, let X be the crosscut, and let Y = 0. The condition X n [x, y] = 0 is satisfied if and only if x ~ y < z for some z E X, or z < x ~ y for some z E X. These possibilities are mutually exclusive, and are in­dicated y < X and X < x, respectively. Thus

2: p,(O,x)C(x,y)p,(y, 1) = "'.YEL;Xn["'.!I]-O

= L p,(0, x) C(x, y) p,(y, 1) + 2: p,(0, x)C(x, y) p,(y, 1) = X.Y;II<X Z,lI;X<z

= 2: !5(O,y)p,(y, 1) + 2: p,(O,x)!5(x,l) = 2p,(0, 1). U;y<X x;X<z

Substitution of this formula into that of Corollary 1 completes the proof. I

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Vol. XIX, 1968 Mobius Inversion in Lattices 603

Corollary 3 (The Complementation Theorem). If s is any fixed element in a finite lattice L, then

~(o, 1) == ~ ~(O,x)C(x,y)~(y, 1) X,YES.L

where s.1 is the set of complements of s in L.

Proof. In Corollary 1 to Theorem 7, let X == Land Y == s.1. Note that

C(X II [x, y], Y) == C([x, y], s.1) == 1

if and only if both x and yare complements of s. If ° == 1, qo - ql + ... == qo == 1. If ° '*' 1, then at most one of 0,1 are in s.1. Assume w.l.o.g. that ° ¢ 8.1. Then a subset A disjoint from s.1 U {a} spans if and only if Au {a} spans. A and Au {a} have cardinalities of opposite parity, so qo - ql + ... == 0. Thus qo - ql + ... == == t5 (0, 1), and the corollary follows. I

5. Combinatorial Geometry. A combinatorial geometry (or simply, a geometry) (e.g.: [4]) is most easily defined in terms of the lattice structure of its flats (closed subgeometries). Such lattices, which are called geometric lattices, have the distinguish­ing characteristic that, for all x, y E L

Y covers x= 3 atom p complementary to x in [0, y].

In this definition, "y covers x" means x < y and x < t :0;; Y => t == y. The com­plementarity condition requires X" P == 0, x v p == y. We shall consider only finite geometric lattices here, so this single property will suffice for a definition. Geometric lattices are consequently relatively-complemented semimodular lattices, generated by atoms, generated by coatoms, and possessed of a well-defined rank A (x) == length of all maximal chains from ° to x. (Note ,1(0) == 0.)

The points of the associated geometry are the atoms of the geometric lattice. Thc lines, planes, ... , of the geometry are the sets of points beneath elements of rank 2, 3, ... , respectively, in the lattice.

Linear graphs give rise to geometries. If G is a linear graph with edge set X and vertex set H, the equivalence relation of path-connection along edges in a subset A ~ X yields a partition nA of the vertex set H into A-path connected components. The map a: A -+ nA is a sup-homomorphism from the Boolean algebra B(X) into the partition lattice P(H). The Galois-closed edge sets, ie: the maximal sets A of each rank ,1(a(A)), form a geometric lattice L(G).

The coboundary operator, defined parenthetically in section 2, maps vq O-chains f: H -+ {a, 1, ... , v - 1} to each coboundary t5/, where q is the number of connected components of G. Colorings, those O-chains which have unequal values on the ends of any edge, correspond to coboundaries which take non-zero values on each edge. The kernel kerg of a coboundary is the set g-l(O), which is necessarily closed. "Ve wish to calculate P(x; v), the number of coboundaries with kernel x and values in the ring {a, 1, ... , v - 1}. The number of v-colorings of the graph is vqp(O; v).

A coboundary is freely-determined by its values on any basis (spanning tree) for the graph, so there are VA(l)-.!(x) v-coboundaries with kernel ;;;;x, for any x E L(G).

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604 H. H. CRAPO

By Mobius inversion on L, there are

p(x; v) = L ,u(x,Y)V.!(l)-.!(y) = Ep( ; v)(x) u;z~"

v-coboundaries with kernel x.

ARCH. MATH.

The polynomial p(O; v) is clearly well-defined for any finite Dedekind lattice L, ie: a lattice satisfying the chain condition, and thus having a rank function A.. p(O; v) has been called the Poincare polynomial of the lattice L. Recent unpublished work tends to establish a relation between Poincare polynomials and general "coloring problems" on such lattices.

Poincare polynomials may be considered as polynomial-valued elements of the incidence algebra of the lattice L. Let

p (x, z; v) = L,u (x, y) C(y, z) VA(z)-.!(y) , 1IeL

so that p(x; v) = p(x, 1; v). Such polynomial. valued matrices have easily-calculable inverses in the incidence algebra.

Theorem 8. Let functions a, b, c on a finite lattice L take value8 which are invertible element8 of a ring. Then

q(x, z) = L a(x),u(x, y)b(y)C(y, z)c(z) 1Ie L

has an inver8e

in the incidence algebra of L. In particular, the Poincare polynomial p(x, z; v) has inver8e

p-l (x, z; v) = L v.! (!I)-.!(:t) ,u(x, y) C(y, z). I 1Ie L

Every geometric lattice L may be realized in a number of ways as a quotient of other geometric lattices with respect to sup-homomorphisms which also pre­serve the relation covers-or-equals. Such maps are called 8trong maps, and map atoms either to atoms or to o. There is a notion of orthogonality [3] ~ith respect to any

such realization (1: M --* L, giving rise to a strong map a*: M --* L*, whenever the domain lattice M is also modular. The relation (1** = a holds.

If the Poincare polynomial is modified so as to have two numerical variables, it becomes possible to obtain from the polynomial for a Boolean representation, by simple substitution of variables, the corresponding polynomial for the orthogonal geometry. For graphs, this process converts coboundary enumeration to cycle enumeration 2). The appropriate two-variable polynomial is the coboundary polynomial for any strong map (1: P --* L, defined by

(6) 1'(a; rp, v) = Lrp.!(ULl(:t»p(x; v). :teL

2) Cf. [2], [7], [8]. "A Ring in Graph Theory" is an important work, in which TUTTE calculates what has come to be known as the Grothendieck group, for a category of graphs.

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Vol. XIX, t 968 Mobius Inversion in Lattices 605

Theorem 9. If a: P --+ L is a strong map between geometric lattices P, L

(7) r (a; cp, v) = L cpA (x) ,u (x, y) V.1(1)-.1(a(x» . X,YEP

Proof. Directly from Theorem 3. I The application of Theorem 9 to Boolean representations of a geometry, such as

the map from subsets A of the set of atoms to sup A in L, is particularly useful.

Corollary 1. If B = B (X) is a finite Boolean algebra and a: B --+ L is a strong map into an geometric lattice L, then

r(a; cp, v) = Lcplxlp(x; v) = L(CP _1)IAl v.1(1)-.1(a(A». I XEL A>;X

The definition of orthogonality relative to a strong map a: M --+ L of a modular geometry onto a geometry L is as follows. The strong map a: M --+ L determines a closure J = aLI (a) on M. There is a unique co closure J* on M satisfying, for all x, y in M such that y covers x

(8) y ;;:;; J (x) ~ J* (y) ;t; x.

The coclosure J* determines a quotient lattice P (with order induced by that on M), and a map r: M --+ P which is an inf-homomorphism preserving the relation covers­

or-equals. The inverted lattice P is a geometric lattice, so we set L* = P. Let

a* be the associated strong map from if onto L*. The rank generating function e of a strong map a: P --+ L defined by

(9) e(a;;, 1]) = L;.1L(l)-.1L(a(x»1].1p (x)-.1L(a(x» XEP

has symmetry [2] relative to orthogonality, whenever a maps a modular geometry onto L.

Theorem 10. e(a*;;, 1]) = e(a; 1],;) for any pair a: M --+ L, a*: M --+ L* of orthogonal maps of a modular geometry.

Proof. The measurement Ad1) - AL(a(x)) which provides the exponent of; in e (a), enumerates the number of intervals [y, z] of length 1 ("steps") in any maximal chain ("path") from x to 1 in M, for which z ;t; J (y), ie: J* (z) ;;:;; y. But this is precisely the measurement AM(X) - AL*(a*(x», which provides the exponent of 1] in e (a*). Similarly,

is the number of steps [y, z] in any path from 0 to x in M for which z ;;:;; J(y), ie: J* (z) ;t; y. I

Corollary 2. If B = B(X) is a finite Boolean algebra and a: B --+ L is a strong map into a geometric lattice L, then

r(a; cp, v) = (cp - 1).1L(1) e C,.:. 1 ' cp - 1).

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606 H.H. CRAPO ARCH. MATH.

Proof. By Corollary 1,

-ria; rp, v) = L (rp - l)IA I V1L(1)-l(u(A» = A!;X

= (rp - 1)ld1) L ( ':'1 )lL(1)-l(U(A»(rp _ l)IAI-l(u(A». I A!;;X rp

We may now complete the calculation of the cycle polynomial -r(a*) from the coboundary polynomial -ria).

Corollary 3. If B = B(X) is a finite Boolean algebra and a: B --+ L is a strong

map onto a geometric lattice L, with orthogonal a*: B --+ L*, then

-r(a*; rp, v) = (rp _ l)lllv-l(1)-r(a; v ~~ ~ 1, v).

Proof. From Corollary 2 we have e(~, 'Yj) = 'Yj-A(1)-r(a; 'Yj + 1, ~'Yj). Also A*(1) = = 111- A(1). Thus

-r(a*; rp, v) = (rp - 1)1 1 1-A(1) 12* C .:. 1 ' rp - 1) = (rp - 1) 111-A(1) 12 (rp - 1, rp':' 1) =

= (rp _ l)III-A(1)V-A(l) (rp _ l)A(1)-r (a; v ~ ~ ~ 1 , v). I

So far we have dealt with representations of geometries as quotients of simpler geometries of higher rank. A few parallel results are available for embeddings of a given geometry as a subgeometry of various larger geometries, usually of equal rank.

Corollary 4. If a: P --+ L is a strong map between geometric lattices and if t: L --+ N is a 1·1 strong map from L into a geometric lattice N, then

-r(da); rp, v) = vAN (1)-AN(,(1»-r(a; rp, v).

Proof. -r(da); rp, v) = L rpAP(UA (x» ,uL(X, y) V·N(l)-AN('(Y» = V1N(1)-lN(,(1» -ria; rp, v) ::z:,yeL

because AN (t (y)) = Ady) for all y E L. I

Corollary 5. If L: L --+ N is a 1-1 strong map from a geometric lattice L into a geo­metric lattice N, then the relation

vlN(I)- Ad 1) PLiO, v) = L PN(x; v). xEN;,A(x)=O

In particular, if N is the lattice of aU partitions of the set H of vertices of a graph G and L is the lattice L(G) of closed subsets of the edge set X of G, then

00

vIHI-I-A(,(X» PLiO, v) = L ndv - 1) (v - 2) ... (v - k + 1) k=l

where nk is the number of k-part color.partitions of the vertex set H of G.

Proof. Evaluate -r(t; rp, v), simplify by using Corollary 4. and set rp = O. I

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Vol. XlX,1968 Mobius Inversion in Lattices 607

Bibliography

[1] H. H. CRAPo, The Mobius Function of a Lattice. J. Combinatorial Theory 1,126-131 (1966). [2] H. H. CRAro, The Tutte Polynomial. Aequationes Math. (to appear). [3] H. H. CRAPO, Geometric Duality. Rend. Sem. Mat. Univ. Padova 38, 23-26 (1967). [4] D. A. HIGGs, Strong Maps of Geometries. J. Combinatorial Theory 6 (1968) (to appear). [5] O. ORE, Galois Connexions. Trans Amer. Math. Soc. 61i, 493-513 (1944). [6] G.-C. ROTA, On the Foundations of Combinatorial Theory I. Z. Wahrscheinlichkeitstheorie

und verw. Gebiete 2, 340-368 (1964). [7] W. T. TUTTE, A Ring in Graph Theory. Proc. Cambridge Philos. Soc. 43, 26-40 (1947). [8] W. T. TUTTE, A Contribution to the Theory of Chromatic Polynomials. Canad. J. Math.

6, 80-91 (1954).

Eingegangen am 9. 11. 1967

AnBchrift des Autora: Henry H. Crapo Department of Mathematics, University of Waterloo. Waterloo, Ontario, Canada

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