i. 2.

3.

LITERATURE CITED

A. A. Bovdi, Group Rings [in Russian], Uzhgorod Univ. (1974). R. Sandling, "Units in the modular group algebra of a finite Abelian p-group," J. Pure Appl. Algebra, 33, 337-346 (1984). S. A. Jennings, "The structure of the group ring of a p-group over a modular field," Trans. Am. Math. Soc., 50, 175-185 (1941).

WIELANDT-TYPE BOUNDS FOR PRIMITIVE MAPPINGS OF PARTIALLY

ORDERED SETS

V. S. Grinberg

i. Introduction. Let N = {i, 2, ..., n}, let X = 2 N be the set of all subsets of the set N. A mapping ~: X--~X is said to be additive if ~ (2") = ~ and ~ (~Jl U ~f2) = (All) U ~ (~f2) for all MI, M 2 e X. Obviously, such a mapping is completely defined

by its values on one-element sets. The additive mappings form a semigroup with identity with respect to composition.

The additive mappings are closely related with nonnegative matrices. This connection arises by associating to an arbitrary nonnegative n x n matrix A = { aij} the additive map- ping ~A: f~A (J) if and only if aij~0. The mapping A ~-, qa is a homomorphism of the semigroup of all nonnegative n x n matrices into the semigroup of the additive mappings of the set X.

Additive mappings have been investigated with various purposes by several authors [i-7]. In this connection, of great importance is the class of the so-called primitive mappings. This concept has arisen in the classical investigations of Perron and Frobenius on nonnegative matrices. Namely, a nonnegative matrix is said to be primitive if some of its natural power is positive. Accordingly, an additive mapping ~ is said to be primi- tive if there exists a natural number t such that ~t(M) =N for all M # ~. The smallest t for which this equality holds will be denoted by t(~).

In [8], Wielandt has given without proof the following result:

t ( ~ ) < n 2 - 2 n + 2 (1)

for any primitive additive mapping ~. A series of examples are also given, showing that the bound (i) is sharp for all n. Various proofs of the inequality (i) can be found in [i, 2, 9].

The above presented situation is generalized in [i0], where for X one considers an arbitrary lattice, while the corresponding class consists of the mappings that preserve sup. (Thus, in the case X = 2 N this class is the class of additive mappings.) Generali- zing in an appropriate manner the concept of a primitive mapping, the author obtains a bound that is similar to (i).

In this paper we consider an even more general situation: for X we take an arbitrary partially ordered set, while the corresponding class consists of the monotone mappings of X into itself. This class is significantly larger than the class of additive mappings (in the case X = 2N). Introducing an appropriate definition of a primitive mapping, we obtain bounds similar to (i) and we prove their sharpness. This gives us the possibility to extend the results, known for nonnegative matrices, to polynomial operators of an arbi- trary degree with nonnegative coefficients.

2. Definitions and Notations. Let X be a finite ordered (partially) set with least element 0 and largest element i. We denote by r the set of all mappings ~: X ~ X, satisfying the conditions:

UkrkommunNllproekt. Translated from Matematicheskie Zametki, Vol. 45, No. 6, pp. 30-35, June, 1989. Original article submitted April 28, 1988.

450 0001-4346/89/4556-0450 $12.50 �9 1989 Plenum Publishing Corporation

i) if x ~ y, then ~ ( x ) ~ (y) (monotonicity);

2) ~ ( o ) = o , ~ ( l ) = l .

Obviously, r is a semigroup with identity with respect to composition of mappings.

A nonnegative integer b is said to be the height of the ordered set X if X contains a chain 0 = x D < x I < ... < x b = i and it does not contain a longer chain of this form. In order to avoid trivialities, we shall assume that b ~ 2.

Let N be the set of the minimal elements in the set obtained from X by removing the zero element. The cardinality of the set N will be denoted by n and we call it the width of the ordered set X.

We define ~E~: ~ (0) = 0, ~(x) = I (x~0). A mapping ~ is said to be primitive if ~ = ~ for some natural number t. The smallest possible t in this equality will be denoted by t(~). Obviously, if ~ is primitive and I .>-t(~), then ~' = ~.

The mapping ~ is said to be indecomposable if from ~ (x)•x there follows that x=0orx=l.

3. Upper Bounds for l(~) . There exists a close connection between primitivity and indecomposability.

THEOREM i. If ~ is primitive, then ~ is indecomposable for all natural numbers j.

Proof. Let ~'(x)<x for some x �9 X and for some natural number j. Then from the monotonicity of ~ there follows that ~ (x)<x for all natural numbers k. If x ~ 0, then ~J(x) = I for kj~t(~), from where x = i.

Theorem 1 admits a converse in the following strengthened form.

THEOREM 2. If for ~ the mappings ~J are indecomposable for j = I, 2 ..... n then ~ is primitive.

Proof. We fix arbitrarily x 0 �9 N. From the indecomposability of ~ there follows that (x)#=0 for x ~ 0. Making use of this, we construct a sequence ~ = (x0, xl, x 2 .... ) of

elements from N, subjected to the condition

x i + l < ~ ( z O , (i = 0 , t , 2 . . . . ). (2)

Assume that the elements x0, x I, ..., xa_ I are pairwise distinct, while xa = x~, c<a. The numbers a = a (~) and c = c(~) depend on the selected sequence and satisfy, obviously, the inequalities 0 ~ c ~ a ~ n. From the monotonicity of @ and from condition (2) there fol- lows that ~ a ~ (~). Applying to this inequality the mapping ~a-c we obtain the chain xc < ~a-c (xc) ~ ~2(a-=) (~) ~ .... The stabilization of this chain occurs in at most b steps, from where ~(~-a)(a-~)(~)= ~(a~)(x~). Now, from the indecomposability of the mapping ~a-= we obtain that ~(0-,)(a-c) (x~) = I, from where x~ < ~c (x0) '

~(~-~)(~-~)+~ (Xo) = 1.

Assume now that x �9 X is an arbitrary element, different from 0. We find x 0 �9 N, for which x 0 ~ x. From (3) there follows that ~(z~)= i for some t. From the monotonicity of ~ there follows that ~'(x) = I. Theorems 1 and 2 have been known for additive mappings [2, 3, 11].

THEOREM 3. Let ~ be a primitive mapping. Then

t (~ ) ~< bn - - n . (4)

Proof. We estimate the exponent in (3):

( b - - 1 ) ( a - - c ) + c = ( b - - 2 ) ( a - - c ) + a < ( b - - 2 ) n + n = b n - - n .

The bound (4) can be strengthened if X satisfies the following condition:

(A) for each x �9 X\(N U {0}) there exist two distinct elements y �9 N and z �9 N such that y < x and z < x.

451

THEOREM 4. Assume that condition (A) is satisfied and ~ is a primitive mapping. Then

t (~) ~< bn - - b - - n -+- 2 . (5)

P r o o f . We show t h a t t h e s e q u e n c e a = (x 0, x~ , x 2, . . . ) , c o n s t r u c t e d i n t h e p r o o f o f Theorem 2, can be subjected in this case to the additional condition

a ( a ) - - c ( a ) ~ < n - - l . ( 6 )

I n d e e d , i f ( 6 ) does n o t h o l d , t h e n n e c e s s a r i l y a ( a ) = n and c ( a ) = O, i . e . , x0 , x 1, ..., Xn_ I are all the elements of the set N and x n = x 0 . From the indecomposability of ~'~ there follows that at least one of the inequalities x~+1 < ~ (xi) (i = 0, I ..... n -- I) is strict, i.e., x~+l< ~ (x~) for some k (0 ~ k ~ n - i). Then from (A) there follows that there exists an index j (i ~ j E n, j # k + i) such that x]< ~(x~). If j < k § i, then for the sequence ~ = (x 0, x I, .... x k, x .- +'i,) we have a(~) = k + I, c (~) =], a(~)-- c(~) : k+l --].~n--l. If, however, j >3K then for the sequence 7 = (x0, xl .... , Xk, Xj, Xj+ I ..... X n .... ) we have

a(?) = n - - / + k + 1, c(?) = 0 , a ( ? ) - - c ( ? ) = n - - ] + k + t ~ < n - - t .

Making u s e o f ( 6 ) , f o r t h e e x p o n e n t i n (3 ) we o b t a i n t h e bound

( b - - t ) ( a - - c ) + c = ( b - - 2 ) ( a - - c ) + a ~ < ( b - - 2 ) ( n - - t ) + n = b n - - b - - n + 2 .

Now we consider the special case when X is the set of all subsets of the set N, ordered by inclusion. The monotonicity of the mapping ~: X-+X means that MI c M 2 implies

( M 1 ) ~ ~ (Mf) ( ~ f , , ~ l ~ X ) . Theorem 4 allows us to generalize the bound (i).

COROLLARY. Let N = {i, 2, .... n}, let X = 2Nm and let T : X + X be a monotone primi-

tive mapping.

Proof. where b = n.

T h e n

t ( q ) ) ~ n 2 - 2 n - + - 2.

Condition (h) is satisfied in this case and,

(7)

therefore, the estimate (5) holds,

4. Sharpness of the Obtained Bounds. The following two theorems show the sharpness

of the inequalities (4) and (5).

THEOREM 5. For any natural numbers b and n there exist an ordered set X with 0 and i, height b, and width n and a primitive mapping ~: X-+X, for which t(~) = bn--n.

Proof. We set X = {0, i} O {xij} (I ~ i ~ b - i, 1 ~ j ~ n). The defining inequali- ties are given in the following manner: xij < xi+x, j (i ~ i ~ b - 2, 1 ~ j ~ n). We set

q~ (xb- l , . ) = t .

(1<i<b--1,1</<n--1), (1 ~<i<b--2) ,

THEOREM 6. For any natural numbers b ~ 2, n e 2 there exist an ordered set X with 0 and I, height b, and width n for which condition (A) is satisfied, and a primitive mapping

~: X + X , such t h a t t (~ ) = b n - - b - - n + 2.

Proof. We set X = {0, i, xlx } U {xij} (2 ~ i ~ b, 2 ~ j ~ n); further, let

Xij < Xi+l, j ( 2 ~ < i ~ < b - - t , 2 ~ < ] ~ < n ) ; x~j'< xi+l,j+l (2 ~< i ~<. b - - l , 2.~< ].<< n - -1 ) ; xi,~ < xi+2,2 (2 ~ i ~< b - - 2); Xll < Xa2 ( fo r b ~ 3).

Finally, we set

(XlO = xf~; ( 2 ~< i ~< b, 2 ~< l ~< n -- t ) ;

452

q~ (xi~) = xi+l,2 (2 ~.< i ~ . b - - t ) ; q~ ( x ~ ) = t .

5. Applications. We apply the obtained results in the investigation of the combina= torial properties of mappings of convex polyhedra into themselves.

Let K be a convex closed polyhedron (not necessarily bounded) in a real space. We introduce on K an equivalence relation, setting x - y if the faces of least dimensions, to which the points x and y belong, coincide. An operator A: K + K issaid to be combina- torial if x ~ y implies Ax ~ Ay. It is easy to verify that, for example, each linear (homo- geneous or nonhomogeneous) operator, preserving K, is combinatorial. According to the introduced equivalence, the factorization of the pair (K, A), where A is a combinatorial operator, yields the pair (K, A), where K is the set of the faces of the polyhedron K, ordered in the natural manner, while A is the quotient operator. Int K is the largest element in K. If in K one has more than one face of minimal dimension s, then it is neces- sar[ to introduce a fictitious face of dimension s - i, which will be the least element in K. If the~quotient operator A is monotone z then the obtained results can be applied to the pair (K, A). For the monotonicity of A it is sufficient, for example, to have the con- tinuity of the operator A.

The following theorem has been given in [i0] without proof.

THEOREM 7. Assume that the unit ball K of an m-dimensional Banach space is a polyhed- ron, having 2n vertices and suppose that the linear operator A satisfies the condition iLAII = 11Amn-m+lil = i. Then iiAJil = i for all natural numbers j.

Proof. Condition licit = 1 means that the operator A maps K into itself. From its linearity there follows that it is combinatorial, i.e., the pair (K, A) generates the pair (K, A). Performing an additional factorization, consisting in the identification of the opposite faces, we obtain the pair (X, q ), to which we apply Theorem 4. The height of X is equal to m + 1 and the width is n; condition (A) is satisfied. If iIAJli < I for some j, then ~ is primitive and, by Theorem 4, ~ =~,~ where k = bn - b - n + 2 = mn - m + i. This means that Ak(K) c Int K, in contradiction with the assumption of the theorem.

The following theorem has been proved in [12] in a rather cumbersome manner. A sim- pler proof has been obtained in [13] on the basis of the general results of [14].

THEOREM 8. Let C m be the m-dimensional arithmetic space, where the norm of a vector is defined as the maximum modulus of its components; let A be a linear operator in Cm, satisfying the condition flail = flAm=-m+11l. Then llA3il = 1 for all natural numbers j.

Proof. We consider the adjoint operator A*. The unit ball of the space Cm* has 2m vertices. Applying Theorem 7 to A* and making use of the fact that iiA*ll = iiAti, we obtain the required result.

In [12] it has been shown that Theorem 8 cannot be improved; namely, an example of an operator A in C m has been constructed for which

I! A I! = t! A . . . . . II = t , b u t II A . . . . "~+~ II < I .

THEOREM 9. Let K be a coordinate cone in the m-dimensional arithmetic space and let A: K + K be a polynomial operator with nonnegative coefficients, preserving 0. Assume that At(K\{0}) c Int K for some natural number t. Then Am2-2m+2(K\{0}) c Int K.

Proof. Under the assumptions of the theorem, the pair (K, A) admits the above des- cribed factorization, yielding the pair (X, q), where X is the set of the faces of the cone K and ~: X--~X is a monotone primitive mapping. The set X is isomorphic in a natural manner to the set of all subsets of the set {i, 2 .... , m}. Applying (7), we obtain the required result.

The author is grateful to Yu. I. Lyubich for his constant interest.

1.

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453

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ii. S. Schwarz, "A new approach to some problems in the theory of nonnegative matrices," Can. J. Math., 16, (91), No. 2, 274-284 (1966).

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13. Yu. I. Lyubich and M. I. Tabachnikov, "On a theorem of Marik and Ptak," Sib. Mat. Zh., i_~0, No. 2, 470-473 (1969).

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