Kříž, Igor

Sgall, Jiří

English

Kříž Igor

Sgall Jiří

University of Szeged

Acta Sci. Math., 55 (1991), 59—65 Well-quasiordering depends on the labels IGOR KRIZ and JIRI SGALL 1. Introduction A well-quasiordering ( W Q O ) is a quasiordered set containing no infinite de-creasing chain and no infinite antichain. A considerable part of the results on W Q O is of the form that a concrete category Q is WQO, where as b means that there exists a Q-morphism from a to b. As an example let us mention the category T of finite trees with tree embeddings (see [2]). The recent solution of the Wagner's conjecture by ROBERTSON and SEYMOUR ([5]) is also of the form that certain category is WQO. Other categories are trivially W Q O , for example the category F of finite sets and injective mappings or the category H of finite linearly ordered sets and strictly increasing mappings. Still, we come to non-trivial questions, if we introduce more involved orderings: Let A be a W Q O and let Q be a concrete category with finite objects and injec-tive morphisms. W e consider a class Q(A) o f objects of Q "labeled by " elements of A at each point. W e put a^b if there is a morphism from a to b which increases the labels (not necessarily strictly). N o w the question is: Is it true that (1) Q(A) is W Q O whenever A is W Q O ? (1 ) w a s p r o v e d f o r F, H b y HIGMAN [1] and f o r T b y NASH-WILLIAMS [4]. O f course, it would be useful if (1) were implied by a simpler condition, say, (2). Q(y) is W Q O for any y^Ord. Although this is not known in general, it was proved recently by one of the auth-ors ([3]) for a considerably broad class of categories ( for all subcategories of H ) . Let us note that, by an easy cardinality argument, (2) is equivalent to (3) , 2 K ) is WQO. Received August 4, 1988. 60 I. Kfiz and J. Sgall So, for subcategories of H, (3)—(1). However, it has not been known if (1) were not implied by a still weaker condition, say (even!), (4) 0 ( 2 ) is WQO. It is the purpose of this paper to present a bunch of counterexamples of this kind. To be exact, we show that the set (5) M = { y| (30 ( ( (V j? < y)Q(fi) is W Q O ) & ( g ( 7 ) is not W Q O ) ) } is confinal in (Mj. W e also prove that M i ® , showing that (4) does not imply Q(3) to be W Q O . 2. Preliminaries 2.1. Conventions and notation. The cardinality of a set X is denoted by \X\. For the ordinals, we use that definition where y is identified with OW < y}-In particular, this will apply to natural numbers. A quasiordering is a reflexive and transitive relation. In a quasiordering, a sequence (at) (finite or infinite) is called bad if / < j - a, $ ai and is called good if ' < 7 - a, = "j-Each infinite sequence contains an infinite good subsequence or an infinite bad sub-sequence (Ramsey theorem). A quasiordering is called W Q O , if no infinite sequence is bad. This definition is equivalent to that used in the Introduction by the Ramsey theorem. For a category C and objects a, b the symbol C(a, b) designates the corres-ponding hom-set and the symbol Ida designates the identity on a. For a concrete category, let the forgetful functor be denoted by U. 2.2. D e f i n i t i o n . In this paper, a QO-category is a concrete category with finite objects and injective morphisms. For a QO-category O and a quasiordering A, put Q(A) = {z = (w., cz)|c, is an object of Q and u.: cz — A). The quasiordering on Q(A) is given by z ^ i if there exists a Q-morphism <p: c, such that u,oU((p)^tiz (pointwise). W e also say that z^t via the mor-phism (p. In the sequel, we shall use the symbol M for the set defined by formula (5) of the Introduction. Well-quasiordering depends on the labels 61 3. The results T o warm up, we start with a special theorem which demostrates the basic idea o f our construction. 3.1. T h e o r e m . M 2 c o . P r o o f . Let a category Qk consist o f finite sets an (n£(o) where each an is a dis-joint union of k sets a°n, Moreover, we shall assume lim |ail = a> for each i£k. n CJ The hom-set Qk(an, am) will be (a) 0 if n >- m, (b ) { I d J if n = m, (c) {<p: an - am| q> injective and ( V / € * ) ( p ( a i ) c (J < ) & (3 ¡ek) (q>(a l n ) £ o j , ) } jSi if n < m. T o see that Qk (k) is not W Q O , let zn (u„, a„) where w„ sends aln to i for each ick. It is easily seen that (z„)„e<3 is a bad sequence. T o prove that Qk(i) is W Q O for i<k, introduce an auxilliary category Qk with the same objects as Qk and with the same morphisms f rom a„ to am for n g m , while for w < m <2kO„, am) = {<p: an - ajcp injective and ( i i£k) (<p(4) £ U <)}. jsi Let (Z,)=((H , , «„(,) ) ) be a bad sequence in Q k ( i ) , i < k . O f course, we have l imn ( i ) = co, « — CO since (z,) is bad. By HIGMAN'S theorem [1], we may assume that (z,) is good with respect to Qk(i). Let, in Qk(i), via cps>t: an(s)-~an(t). N o w , since i<k, there is a such that, for each Kdco, there exist p, r£k, p>r, and t(K)£a> such that ({*€ap(r(K» I ut{K)(x) = j}\ s K, |{-^€<(t(K))I ut(K)(x) = /}| s K. Without loss o f generality, we may assume p, r fixed and / (0 )=0. Put t = t{\tfnW\ +-1). Thus, there exist y£ar0 such that u0(x) = u,(y) =y Im <p0.,. 62 I. Kriz and J. Sgall W e conclude that, in Qk(i), z0^z, via a mapping cp given by cp(z) = <p0,,(z) if z x <P(x) = y, contradicting our assumption. 3.2. T h e o r e m . M is confinal in (o^. P r o o f below in 3.8. 3.3. The Constructions. Let t o ^ y ^ a ^ . The there exists a bijection sy: o)--y. Let the objects of Cy be the sets ay(n) = {iy(0),..., jy(n-l)}. The hom-set Cy(ay(k), ay(n)) will be (1) 0 for k > n , (2) {IdDy(n ) } for k = n, (3) the set of all injective mappings (p: ay(k)^ay(n) such that, for some j<k, ( a ) <p{s-,(j)) < sy(j), (b) for i < j, (p{sy{ij) = sy{i), (c) for i rSj, 0 < sy(i) - (p (fi) < sy(i). 3.4. L e m m a . Cy is a QO-category. P r o o f . What remains to show is that, for k < m < n , cp£Cy(ay(k), ay(n)), \jj£Cy(ay(m), ay(n)) \l/o(p£Cy(ay(k), ay(n)). To this end, let <p, tp satisfy the statement of (3) with constants j^k, ] < m , respect-ively. W e will show that ij/o(p satisfies it with the constant min ( j , ]). W e distin-guish two cases: Case 1. J =./: The proof of (a), (b), (c) for ij/o<p is contained in the following computations. (By (a) for cp, ] and (c) for \p) \po(p(sy(j)) < sy(j). For /<7 (by (b) for <p, if/ and j ^ f ) $o<p(Sy(i)) = \j>(sy(ij) = sy(i). For /Ha./, P<sy(i), (by (c) for <p, i¡j and ^ocp(P)^ sy(i). Well-quasiordering depends on the labels 63 Case 2. ]<j: Compute again. (By (b) for <p and by (a) for \jj) lj/0(p(sy(])) = ip(sy(j)) = sy(J). For i'-=j, (by (b) for <p, ij/) \jjO(p (Sy (/)) = I¡/(sy(i)) = sy(i). For isaj, P~=sy(i), (by (c) for <p, ip) . ^ocp(P) < sy(i). 3.5. L e m m a . Cy(y) is not WQO. P r o o f . Introduce a sequence z„=(u„, c„) in C y ( y ) by putting cn = ay(n) u„{sy(i)) = sy(i). By condition (a) in 3.3 (3) we see easily that (z„)„€(0 is bad in cy(y). 3.6. A u x i l l i a r y d e f i n i t i o n . Let us call a pair {fi, a), a, P^.o)1 admissible, if there exist a y = cc, an increasing sequence (N(Z))IG(0, a number K£CD and a bad sequence z,-= («,-, ay(n(i))) in C-^cOy) such that for each i£co |{<5 < a|5eay(w(z))&w ;(5) p}\ < K. 3.7. L e m m a . (1) If Cy(fi) is not W Q O then (fi, y) is admissible. (2) If {fi, a) is admissible and a < a then (ft, a) is admissible. (3) (0, co) is not admissible. (4) I f ( f i , a + co) is admissible then there exists a P such that (p, a) is admissible. P r o o f . (2) and (3) are obvious. Note that in (1) we may use K= 1. W e shall prove (4). Consider the entities y,n(i), K,zt constituting the admissibility of (P,a + (o). Let p = max {/|a = sy(i) ----- <x-\-K). Further, let a p + 1—y, K ( 0 ) , = {a0 < < . . . < ap }. For i£p+1, t£co, define ( « ] ) (recall that F is the category of finite sets and injective mappings) in the following way: £¡(0 = K ( ./ ' ) !7 < n{t), ctf < .y./y) < a i + 1 } , bM = ( « ¡ k , ( 0 . C;(0). By Ramsey and Higman's theorems there exists an increasing sequence (tx)xia> 64 I. Kriz and J. Sgall such that n(tx)>p and for x^y, i£p+1 (*) «,>,) ^ u, {a,) ( ** ) bi(tx) ^ bi(ty) in FCcoj) via some mapping (pi(tx, ty). Without loss of generality, we may assume tx=x. Now, by ( * ) and by the definition o f K there exists an a § a < a + K such that for all t ",(«) < P. We will show that (w0(a), a) is admissible, concluding the proof of (4) by (2). In fact, for the new K we may take « (0 ) : If, for some t > 0 , there are more than « (0 ) , of y '< « (/ ) with ut(ny(j))^u0(a) then there exists at least one such /' that neither j^p nor sy(J) lies in the image of <p;(0, 0 for any i. N o w define cp: ay(n(0)) — - f l r ( « ( 0 ) by (p(sy(ij) = sy(t) whenever i£p+], s(i) ^ a tp( a) = sy(j) <p(5) = <p;(0, t)(6) if 5£Ci(0). W e see easily that <p£Cy(ay(n(0)), ay(n(t))) and z0 ^ z, via (p, contradicting the assumption that (z,) is bad. 3.8. P r o o f o f T h e o r e m 3.2. Define y(fi) inductively by 7(0) = (0 y(P + l) = y(P)+co y(p) = (limyO?,))+co for pyp. I t follows from 3.7. (2), (4) that (p, y(fij) is not admissible for any p. Thus, by 3.7. (1), cv(fi) ifi ) is WQO. This, together with Lemma 3.5, concludes the proof of Theorem 3.2. References [1] G. HIGMAN, Ordering by divisibility in abstract algebras, Proc. London Math. Soc., (J) 2 (1952), 326—336. [2] J. KRUSKAL, Well-quasi-ordering, the three theorem, and Vazsony's conjecture, Trans. Amer. Math. Soc., 95 (1960), 210—225. Well-quasiordering depends on the labels 65 [3] I. KRJ'Z, and R . THOMAS, On well-quasi-ordering finite structures with labels, Graphs and Combinatorics, 6 (1990), 41—44. [4] C. ST. J. A . NASH-WILLIAMS, On well-quasi-ordering finite trees, Proc. Camb. Phil. Soc., 59 (1963), 833—835. [5] N. ROBERTSON and P. D . SEYMOUR; Graph minors I—Xm, submitted (1986—1988). a. K.) DEPT. OF APPLIED MATHEMATICS UNIVERSITY OF CHICAGO CHICAGO, IL 60637 (J.S) SCHOOL OF COMPUTER SCIENCE CARNEGIE MELLOW UNIVERSITY PITTSBURGH, PA 15213—3890 5 

Well-quasiordering depends on the labels

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Well-quasiordering depends on the labels

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