aa r X i v : . [ m a t h . L O ] M a y AN INFINITE NATURAL PRODUCT
PAOLO LIPPARINI
Abstract.
We study an infinite countable iteration of the nat-ural product between ordinals. We present an “effective” way tocompute this countable natural product; in the non trivial casesthe result depends only on the natural sum of the degrees of thefactors, where the degree of a nonzero ordinal is the largest expo-nent in its Cantor normal form representation. Thus we are ableto lift former results about infinitary sums to infinitary products.Finally, we provide an order-theoretical characterization of the in-finite natural product; this characterization merges in a nontrivialway a theorem by Carruth describing the natural product of twoordinals and a known description of the ordinal product of a pos-sibly infinite sequence of ordinals. Introduction
The usual addition ` and multiplication ¨ between ordinals can bedefined by transfinite recursion and have a clear order-theoretical mean-ing; for example, α ` β is the order-type of a copy of α to which a copyof β is added at the top. However, ` and ¨ have very poor algebraicproperties; though both are associative, they are neither commutativenor cancellative, left distributivity fails, etc. See, e. g., Bachmann [B],Hausdorff [Hau] and Sierpi´nski [S3] for full details.In certain cases it is useful to consider the so-called natural opera-tions and b . These operations are defined by expressing the operandsin Cantor normal form and, roughly, treating the expressions as if theywere polynomials in ω . The natural operations have the advantage ofsatisfying good algebraic properties; moreover, in a few cases they havefound mathematical applications even outside logic. See e. g., Carruth[Ca] and Toulmin [T]. Further references, including some variants andhistorical remarks, can be found in Altman [A] and Ehrlich [E, pp.24–25]. It is also interesting to observe that the natural operations Mathematics Subject Classification.
Key words and phrases.
Ordinal number; (infinite) (natural) product, sum; (lo-cally) finitely Carruth extension.Work performed under the auspices of G.N.S.A.G.A. Work partially supportedby PRIN 2012 “Logica, Modelli e Insiemi”. are the restriction to the ordinals of the surreal operations on Conway“Numbers” [Co]. Limited to the case of the ordinal natural sum, thecorresponding two-arguments recursive definition appears implicitly asearly as in the proof of de Jongh and Parikh [dJP, Theorem 3.4].An infinitary generalization of the natural sum in which the supre-mum is taken at the limit stage has been considered in Wang [W] andV¨a¨an¨anen and Wang [VW] with applications to infinitary logics. Thisinfinite natural sum has been studied in detail in [L1], where an order-theoretical characterization has been provided. In the present note weintroduce and study the analogously defined infinitary product. Thecomputation of the infinite natural product can be reduced to the com-putation of some—possibly infinite—natural sum; in particular, we candirectly transfer results from sums to products, rather than repeatingessentially the same arguments. Curiously, the method applies to theusual infinitary operations, too. See Theorem 2.8 and Corollary 2.9;we are not aware of previous uses of this technique.Order-theoretical characterizations of the finitary natural operationshave been provided in Carruth [Ca]. Some characterizations seem tohave been independently rediscovered many times, e. g., Toulmin [T]and de Jongh and Parikh [dJP]. As we showed in [L1], Carruth charac-terization of the finite natural sum cannot be generalized as it stands tothe infinitary natural sum; see, in particular, the comments at the be-ginning of [L1, Section 4]. A similar situation occurs for the infinitarynatural product; see Subsection 3.4 below. In the case of the infinitenatural sum the difficulty can be circumvented by imposing a finitenesscondition to a Carruth-like description: see [L1, Theorem 4.7]. As weshall show in Section 3, a similar result holds for the infinite naturalproduct, though the situation gets technically more involved.To explain our construction in some detail, it is well known thatthe usual finite ordinal product is order-theoretically characterized bytaking the reverse lexicographical order on the product of the factors.Less known, the same holds in the infinite case, too, provided that in theproduct one takes into account only elements with finite support. SeeHausdorff [Hau, §
16] and Matsuzaka [Ma]. We show that the infinitenatural product can be characterized by merging the representation byCarruth and the just mentioned one; roughly, by using Carruth orderfor a sufficiently large but finite product of factors and then working asin an ordinary product when we approach infinity. Quite surprisingly,a local version of the result holds. Namely, if ď is a linear order on thefinite-support-product of a sequence p α i q i ă ω of ordinals and ď is suchthat, for every element c , the set of the ď predecessors of c is built in away similar to above, then ď is a well-order of order type less than or N INFINITE NATURAL PRODUCT 3 equal to the infinite natural product of the α i ’s. Moreover, the infinitenatural product is actually the maximum of the ordinals obtained thisway, that is, the order-type of the product can always be realized inthe above way. See Section 3 and in particular Theorem 3.1 for fulldetails.1.1. Preliminaries.
We assume familiarity with the basic theory ofordinal numbers. Unexplained notions and notations are standard andcan be found, e. g., in the mentioned books [B, Hau, S3]. Notice thathere sums, products and exponentiations are always considered in theordinal sense. The usual ordinal product of two ordinals α and β isdenoted by αβ or sometimes α ¨ β for clarity. The classical infinitaryordinal product is denoted by ś i ă ω α i . Recall that every nonzero or-dinal α can be expressed in Cantor normal form in a unique way asfollows α “ ω ξ k r k ` ω ξ k ´ r k ´ ` ¨ ¨ ¨ ` ω ξ r ` ω ξ r for some integers k ě r k , . . . , r ą ξ k ą ξ k ´ ą ¨ ¨ ¨ ą ξ ą ξ . If β is another ordinal expressed in Cantor normal form, the natural sum α β is obtained by summing the two expressions as ifthey were polynomials in ω . The natural product α b β is computed byusing the rule ω ξ b ω η “ ω ξ η and then expanding again by “linearity”.Both and b are commutative, associative and cancellative (exceptfor multiplication by 0, of course). If p α i q i ă ω is a sequence of ordinals, i ă ω α i “ sup n ă ω p α α α n ´ q . See [W, VW, L1] for furtherdetails about .2. An infinite natural product
Definition 2.1.
Suppose that p α i q i ă ω is a sequence of ordinals.For i ă ω , let P i denote the partial natural product α b α b¨ ¨ ¨b α i ´ ,with the convention P “  i ă ω α i “ lim i ă ω P i .This means that  i ă ω α i “ α i is 0 and  i ă ω α i “ sup i ă ω P i if each α i is different from 0 (cf. Clause (5) in Proposition2.2 below).From now on, when not otherwise specified, p α i q i ă ω is a fixed se-quence of ordinals and the partial natural products P i are always com-puted as above and with respect to the sequence p α i q i ă ω .We now state some simple facts about the infinitary operation  .Only (2), (5) and (6) will be used in what follows. Proposition 2.2.
Let α i , β i be two sequences of ordinals and n, m ă ω .(1) ś i ă ω α i ď  i ă ω α i PAOLO LIPPARINI (2) If β i ď α i , for every i ă ω , then  i ă ω β i ď  i ă ω α i (3) If π is a permutation of ω , then  i ă ω α i “  i ă ω α π p i q (4) More generally, suppose that p F h q h ă ω is a partition of ω intofinite subsets, say, F h “ t j , . . . , j r p h q u , for every h P ω . Then â i ă ω α i “ â h ă ω â j P F h α j “ â h ă ω p α j b α j b ¨ ¨ ¨ b α j r p h q q Suppose in addition that α i ‰ , for every i ă ω .(5) If i ă j , then P i ď P j ; equality holds if and only if α i “ ¨ ¨ ¨ “ α j ´ “ .(6) For every n ă ω , we have P n ď  i ă ω α i ; equality holds if andonly if α i “ , for every i ě n .Proof. Easy, using the properties of the finitary b . We just commenton (3). This is trivial if at least one α i is 0. Otherwise, every partialproduct on the left is bounded by some sufficiently long partial producton the right (by monotonicity, associativity and commutativity of b ,and since all factors are nonzero by assumption); the converse holds aswell, hence the infinitary products are equal.Let us mention that (3) and (4) can be also proved by using The-orem 2.6 below and the corresponding properties of given in [L1,Proposition 2.4(5)(6)]. (cid:3) Lemma 2.3. If p β i q i ă ω is a sequence of ordinals and β “ i ă ω β i ,then â i ă ω ω β i “ ω β Proof.
Since ω β i is always different from 0, we have  i ă ω α i “ sup i ă ω P i .Here, of course, P i is computed with respect to the sequence given by α i “ ω β i , for i ă ω .By a property of the natural product (or the definition, if you like),we have P i “ ω β b ω β b ¨ ¨ ¨ b ω β i ´ “ ω β β β i ´ , for every i ă ω . Letting B i “ β β β i ´ , we have by definition β “ i ă ω β i “ sup i ă ω B i . But then  i ă ω α i “ sup i ă ω P i “ sup i ă ω ω B i “ ω β by continuity of the exponentiation. (cid:3) Let α be a nonzero ordinal expressed as ω ξ k r k ` ¨ ¨ ¨ ` ω ξ r in Cantornormal form. The ordinal d p α q “ ξ k will be called the degree or the largest exponent of α . The ordinal m p α q “ ω ξ k r k will be called the leading monomial of α . By convention, we set m p q “
0. The followinglemma is trivial, but it will be useful in many situations.
Lemma 2.4.
For every ordinal α , m p α q ď α ď m p α q ` m p α q “ m p α q ¨ ď m p α q b N INFINITE NATURAL PRODUCT 5
Lemma 2.5. If p α i q i ă ω is a sequence of ordinals which is not eventually , then â i ă ω α i “ â i ă ω m p α i q Proof.
This is trivial if at least one α i is 0.Hence suppose that α i ‰
0, for all i ă ω , and that p α i q i ă ω is noteventually 1. The inequality  i ă ω α i ě  i ă ω m p α i q is trivial by mono-tonicity (Proposition 2.2(2)), since α ě m p α q , for every ordinal α .To prove the converse, we show that, for every h ă ω , there is k ă ω such that  i ă h α i ď  i ă k m p α i q . This is enough since if all the α i ’s arenonzero, then the succession of the partial products is nondecreasing(Proposition 2.2(5)). It is a trivial property of the finitary naturalproduct that m p  i ă h α i q “  i ă h m p α i q (by strict monotonicity of ).Since p α i q i ă ω is not eventually 1, there is k ą h such that α k ´ ě
2. Then  i ă k m p α i q ě ` i ă k ´ m p α i q ˘ b ě p  i ă h m p α i qq b “ m p  i ă h α i q b ě  i ă h α i , by Lemma 2.4. (cid:3) Theorem 2.6.
Let p α i q i ă ω be a sequence of ordinals and let β “ i ă ω d p α i q . The infinite natural product  i ă ω α i can be computed ac-cording to the following rules. â i ă ω α i “ if (and only if ) at least one α i is equal to â i ă ω α i “ α b ¨ ¨ ¨ b α n ´ if α i “ , for every i ě n ;(2) â i ă ω α i “ ω d p α q d p α n ´ q` “ ω β ` if α i ‰ , for all i ă ω , α i ă ω , for all i ě n , and the sequence is not eventually ; (3) â i ă ω α i “ â i ă ω ω d p α i q “ ω β if none of the above cases applies, (4) that is, no element of the sequence is and the members ofthe sequence are not eventually ă ω . Before proving Theorem 2.6, we notice that it gives an effective wayto compute  i ă ω α i , for every sequence p α i q i ă ω of ordinals. Apply (1)if at least one α i is equal to 0; if this is not the case, apply (2) ifthe α i are eventually 1; if not, then exactly one of (3) or (4) occurs.Notice that conditions (1) and (2) in Theorem 2.6 might overlap, and n in (2) and (3) is not uniquely defined, but the conditions give thesame outcome in any overlapping case. Notice also that the expression d p α q d p α n ´ q ` p d p α q d p α n ´ qq ` “ d p α q d p α n ´ q ` q . PAOLO LIPPARINI
Proof.
The result follows trivially from the definitions if some α i isequal to 0 or when the sequence is eventually 1.If we are in the case given by (3), then, for every ℓ ă ω , there is h ą n such that there are at least ℓ -many α i ě
2, where the index i varies between n and h . Thus  i ă h ` α i ě ω d p α q b ¨ ¨ ¨ b ω d p α n ´ q b ℓ since α ě ω d p α q , for every nonzero ordinal α . Since ℓ is arbitrary, we get  i ă ω α i “ sup h ă ω  i ă h ` α i ě sup ℓ ă ω p ω d p α q b ¨ ¨ ¨ b ω d p α n ´ q b ℓ q “ sup ℓ ă ω ω d p α q d p α n ´ q ℓ “ ω d p α q d p α n ´ q` , since ω ε b “ ω ε p ă ω ω ε p “ ω ε ` , for every ordinal ε .In the other direction, we have  i ă ω α i “  i ă ω m p α i q from Lemma2.5, hence it is enough to prove  i ă ω m p α i q ď ω d p α q d p α n ´ q` . If h ă ω , and, for every i ă ω , letting s i be the only natural number suchthat m p α i q “ ω d p α i q s i , then by associativity and commutativity of b ,we get  i ă h ` m p α i q “ ω d p α q b s b ¨ ¨ ¨ b ω d p α h q b s h “ ω d p α q b ¨ ¨ ¨ b ω d p α h q b s b ¨ ¨ ¨ b s h ď ω d p α q b ¨ ¨ ¨ b ω d p α n ´ q b ω “ ω d p α q d p α n ´ q` ,since d p α i q “
0, for i ě n and, by construction, s i ă ω , for every i ă ω .Hence  i ă ω m p α i q “ sup h ă ω  i ă h m p α i q ď ω d p α q d p α n ´ q` .The last identity in (3) follows from the already mentioned fact that d p α i q “
0, for i ě n , hence β “ i ă ω d p α i q “ i ă n d p α i q .The case given by (4) is similar and somewhat easier. The inequality  i ă ω α i ě  i ă ω ω d p α i q is trivial by monotonicity.For the converse, we use again the identity  i ă ω α i “  i ă ω m p α i q from Lemma 2.5. Arguing as in case (3), we have that, for every h ă ω ,  i ď h m p α i q ď ω d p α q b ¨ ¨ ¨ b ω d p α h q b ω , but there is some k ą h suchthat α k ě ω , since the members of the sequence are not eventually ă ω .Hence  i ď h m p α i q ď ω d p α q b ¨ ¨ ¨ b ω d p α h q b ω ď ω d p α q b ¨ ¨ ¨ b ω d p α h q b¨ ¨ ¨ b ω d p α k q ď  i ă ω ω d p α i q . In conclusion,  i ă ω α i “  i ă ω m p α i q “ sup h ă ω  i ď h m p α i q ď  i ă ω ω d p α i q .The last identity is from Lemma 2.3. (cid:3) Corollary 5.1 in [L1] can be used to provide a more precise evaluationof β in case (4) in Theorem 2.6. Corollary 2.7.
Suppose that p α i q i ă ω is a sequence of ordinals such thatno element of the sequence is and the members of the sequence arenot eventually ă ω (thus the sequence p d p α i qq i ă ω is not eventually ).Let ξ be the smallest ordinal such that t i ă ω | d p α i q ě ω ξ u is finite,and enumerate those α i ’s such that d p α i q ě ω ξ as α i , . . . , α i k (thesequence might be empty). Then  i ă ω α i “ ω β , where β “ p d p α i q d p α i k qq ` ω ξ . We need the results analogous to Theorem 2.6 for the classical ordinalproduct.
N INFINITE NATURAL PRODUCT 7
Theorem 2.8. (1) If p β i q i ă ω is a sequence of ordinals and β “ ř i ă ω β i , then ź i ă ω ω β i “ ω β (2) If p α i q i ă ω is a sequence of ordinals which is not eventually ,then ź i ă ω α i “ ź i ă ω m p α i q (3) Suppose that p α i q i ă ω is a sequence of nonzero ordinals which isnot eventually and let β “ ř i ă ω d p α i q . Then ź i ă ω α i “ ω d p α q`¨¨¨` d p α n ´ q` “ ω β ` if α i ă ω , for all i ě n ź i ă ω α i “ ź i ă ω ω d p α i q “ ω β if the sequence is not eventually ă ω Proof. (1) Like the proof of Lemma 2.3, using the identity ω β ω β . . .ω β i ´ “ ω β ` β `¨¨¨` β i ´ .(2) Like the proof of Lemma 2.5. In fact, we do have m p ś i ă h α i q “ ś i ă h m p α i q and this is enough for the proof.(3) is proved as Theorem 2.6. (cid:3) Theorems 2.6 and 2.8 can be used to “lift” some results from infini-tary sums to infinitary products. To show how the method works, wefirst present a simple example, though only feebly connected with therest of this note.Sierpi´nski [S1] showed that a sum ř i ă ω α i of ordinals can assumeonly finitely many values, by permuting the α i ’s. A proof can be foundalso in [L1]. Then Sierpi´nski in [S2] showed the analogous result foran infinite product. Using Theorem 2.8 we show that the result aboutproducts is immediate from the result about sums. Corollary 2.9. If p α i q i ă ω is a sequence of ordinals, one obtains only afinite number of ordinals by considering all products of the form ś i P I γ i ,where p γ i q i ă ω is a permutation of p α i q i ă ω , that is, there exists a bijec-tion π : ω Ñ ω such that γ i “ α π p i q for every i ă ω .Proof. This is trivial if some α i is 0, or if the α i ’s are eventually 1, solet us assume that none of the above cases occurs.If the sequence is eventually ă ω , then the first equation in Theorem2.8(3) shows that we obtain only a finite number of products by takingrearrangements of the factors, since the resulting products are givenby ω δ ` , where δ is a sum of d p α q , . . . , d p α n ´ q , taken in some order,but there is only a finite number of rearrangements of this finite set, PAOLO LIPPARINI hence there are only a finite number of possibilities for δ (notice thatif α i ă ω , then d p α i q “
0, hence the degrees of finite ordinals do notcontribute to the sum).In the remaining case, the products obtained by rearrangements havethe form ω δ , with δ “ ř i ă ω d p γ i q , by the second equation in Theorem2.8(3). The quoted result from [S1] shows that we have only a finitenumber of possibilities for δ , hence there are only a finite number ofpossibilities for the values of the rearranged products. (cid:3) We now use Theorems 2.6 and 2.8 in a slightly more involved situa-tion in order to transfer some results from [L1] about infinite naturalsums to results about infinite natural products.
Corollary 2.10.
For every sequence p α i q i ă ω of ordinals there is m ă ω such that, for every n ě m , â n ď i ă ω α i “ ź n ď i ă ω α i and (5) â i ă ω α i “ p α b ¨ ¨ ¨ b α n ´ q ¨ â n ď i ă ω α i “ p α b ¨ ¨ ¨ b α n ´ q ¨ ź n ď i ă ω α i (6) and if, moreover, every α i is nonzero and the sequence is not eventually , then â i ă ω α i “ ω β β n ´ ¨ â n ď i ă ω α i “ ω β β n ´ ¨ ź n ď i ă ω α i (7) where β i “ d p α i q , for every i ă ω .Proof. The result is trivial if the sequence is eventually 1; moreover,(6) is trivial if some α i is 0. Furthermore, (5) is trivial if, for every i ă ω , there is j ą i such that α j “
0. Otherwise, by taking m largeenough, we have α i ą
0, for i ą m . Henceforth it is enough to provethe result in the case when the sequence is not eventually 1 and all the α i ’s are nonzero.We shall first prove (5) and (7) and then derive (6). If the sequenceis eventually ă ω , then (5) is trivial, since in this case, for large enough n , both sides are equal to ω . Then (7) is immediate from equation (3)in Theorem 2.6, since ω β β n ´ ω “ ω β β n ´ ` .Suppose now that the sequence p α i q i ă ω is not eventually ă ω , hencethe sequence p β i q i ă ω is not eventually 0. By [L1, Theorem 3.1], thereis m ă ω such that, for every n ě m , we have n ď i ă ω β i “ ř n ď i ă ω β i . N INFINITE NATURAL PRODUCT 9
Fixing some n ě m and letting β “ n ď i ă ω β i , we get  n ď i ă ω α i “ ω β “ ś n ď i ă ω α i from, respectively, equation (4) in Theorem 2.6 andthe last equation in Theorem 2.8(3). This proves (5).Letting β “ i ă ω β i , we notice that in [L1, Theorem 3.1] it has alsobeen proved that β “ p β β n ´ q ` β . Using the above identityand applying equation (4) in Theorem 2.6 twice, we get  i ă ω α i “ ω β “ ω β β n ´ ω β “ ω β β n ´  n ď i ă ω α i , that is, (7) (the secondidentity in (7) could be proved in the same way, but now it followsimmediately from (5)).Equation (6) remains to be proved. We shall prove that in the non-trivial cases the expressions given by (6) and (7) are equal. One direc-tion is trivial, since ω β β n ´ “ ω β b¨ ¨ ¨b ω β n ´ ď α b¨ ¨ ¨b α n ´ . Forthe other direction, let us observe that, in the nontrivial cases, againby Theorem 2.6,  n ď i ă ω α i has the form ω β , for some β ě
1. Then p α b¨ ¨ ¨b α n ´ q¨ Â n ď i ă ω α i “ p α b¨ ¨ ¨b α n ´ q¨ ω β ď m p α b¨ ¨ ¨b α n ´ q¨ ¨ ω β “ p m p α qb¨ ¨ ¨b m p α n ´ qq¨ ω β “ p ω d p α q s b¨ ¨ ¨b ω d p α n ´ q s n ´ q¨ ω β “p ω β b ¨ ¨ ¨ b ω β n ´ b s o . . . s n ´ q ¨ ω β “ ω β β n ´ ¨ s o . . . s n ´ ¨ ω β “ ω β β n ´ ¨ ω β “ ω β β n ´ ¨ Â n ď i ă ω α i , where we used Lemma 2.4and the facts that kω β “ ω β , whenever k ă ω and β ě
1, and that ω ξ b k “ ω ξ ¨ k , for all ordinals k ă ω and ξ . (cid:3) An order-theoretical characterization p P, ďq simply as P . However, in manysituations, we shall have several different orderings on the same set; inthat case we shall explicitly indicate the order. It is sometimes conve-nient to define ď in terms of the associated ă relation and conversely.As a standard convention, a ď b is equivalent to “either a “ b or a ă b ”(strict disjunction). We shall be quite informal about the distinctionand we shall use either ď or ă case by case according to convenience,even when we are dealing with (essentially) the same order.In order to avoid notational ambiguity, let us denote the cartesianproduct of a family p A i q i P I of sets by Ś i P I A i . If each A i is an orderedset, with the order denoted by ď i , then a partial order ď ˆ can bedefined on Ś i P I A i componentwise. Namely, we put a ď ˆ b if and onlyif a i ď i b i , for every i P I . Apparently, for our purposes, the abovedefinition has little use when dealing with ordinals (more precisely,order-types of well-ordered sets), since ordinals are linearly ordered,but generally the above construction furnishes only a partially orderedset. However, see below for uses of the ordered set p Ś i P I A i , ď ˆ q . p I, ď I q is reverse-well-orderedand each A i is an ordered set, then the anti-lexicographic order L ˚ i P I A i “p L ˚ i P I A i , ď ˚ q on the set Ś i P I A i is obtained by putting a ď ˚ b if and only if either a “ b , or a i ă i b i , where i is thelargest element of I such that a i ‰ b i .In other words, ď ˚ orders L ˚ i P I A i by the last difference. The definitionmakes sense, since I is reverse-well-ordered. It turns out that if each A i is linearly ordered, then L ˚ i P I A i is linearly ordered and if in addition I is finite, then L ˚ i P I A i is well-ordered. Moreover, for two ordinals α and α , it happens that α α is exactly the order-type of L ˚ i ă α i . Thiscan be obviously generalized to finite products.A similar characterization can be given for infinite products, butsome details should be made precise. Suppose that each A i is anordered set with order ď i and with a specified element 0 i P A i . If a P Ś i P I A i , the support supp p a q of a is the set t i P I | a i ‰ i u .Let Ś i P I A i be the subset of Ś i P I A i consisting of those elements withfinite support. Of course, Ś i P I A i inherits a partial order ď ˆ as a sub-order of p Ś i P I A i , ď ˆ q . If I is linearly ordered, we can consider anotherorder L i P I A i “ p L i P I A i , ď L q on the set Ś i P I A i defined as follows.(*) a ď L b if and only if either a “ b , or a i ă i b i , where i is thelargest element of supp p a q Y supp p b q such that a i ‰ b i .The definition makes sense, since supp p a q Y supp p b q is finite and I islinearly ordered. Of course, when I is finite, Ś i P I A i and Ś i P I A i arethe same set and L i P I and L ˚ i P I are the same order. It is known that,for every sequence p α i q i ă δ of ordinals, L i ă δ α i is well-ordered and hasorder-type ś i ă δ α i . Here the specified element 0 i is always chosen tobe the ordinal 0. See [B, III, §
10 1.2 and §
11 Satz 7], Hausdorff [Hau, §
16] and Matsuzaka [Ma] for details. Of course, we could have seenjust by cardinality considerations that Ś i ă δ A i does not work in orderto obtain an order theoretical characterization of ś i ă δ α i in the casewhen δ is infinite.3.3. Dealing now with natural products, a characterization in the fi-nite case has been found by Carruth [Ca]. He proved that α b α is the largest ordinal which is the order-type of some linear extensionof the componentwise order on α ˆ α . Here α ˆ α is ordered as p Ś i ă α i , ď ˆ q in the above notation. Carruth result includes the proofthat such an ordinal exists, that is, that each such linear extension is awell-order and that the set of order-types of such linear extensions hasa maximum, not just a supremum. From the modern point of view, this N INFINITE NATURAL PRODUCT 11 can be seen as a special case of theorems by Wolk [Wo] and de Jongh,Parikh [dJP], since the product of two well quasi-orders (in particular,two well-orders) is still a well quasi order. See, e. g., [Mi]. Carruthresult obviously extends to the case of any finite number of factors.3.4. Some difficulties are encountered when trying to unify the resultsrecalled in 3.2 and 3.3. Let us limit ourselves to the simplest infinitecase of an ω -indexed sequence, which is the main theme of the presentnote. It would be natural to consider the supremum of the order-typesof well-ordered linear extensions of the restriction ď ˆ of ď ˆ to Ś i ă ω α i .However, just considering Ś i ă ω
2, we see that p Ś i ă ω , ď ˆ q has linearextensions which are not well-ordered. Indeed, for every j ă ω , let b j P Ś i ă ω b jj “ b ji “ i ‰ j . Then the b j ’sform a countable set of pairwise ď ˆ -incomparable elements of Ś i ă ω Ś i ă ω Ś i ă ω
2, we get fromthe above considerations that the supremum of their order-types is ω ,hence this supremum is not a maximum and anyway it is too large tohave the intended meaning, that is, ω “  i ă ω p α i q i ă ω is a sequence of ordinals, then Ś i ă ω α i is theset of the sequences with finite support and the (partial) order ď ˆ isdefined componentwise. If a, b P Ś i ă ω α i and a ‰ b , let diff p a, b q bethe largest element i of supp p a q Y supp p b q such that a i ‰ b i . Thus (*)above introduces a linear order ă L on Ś i ă ω α i defined by a ă L b if andonly if a i ă b i , for i “ diff p a, b q (here ă is the standard order on α i ,hence there is no need to explicitly indicate the index). By the resultsrecalled in Subsection 3.2, the above order ă L has type ś i ă ω α i . Onthe other hand, in the finite case, by Carruth theorem mentioned in3.3,  i ă n α i is the order-type of the largest linear extension of ď ˆ on Ś i ă n α i “ Ś i ă n α i . We show that in the infinite case  i ă ω α i can beevaluated by combining the above constructions.If a P Ś i ă ω α i is a sequence and n ă ω , let a æ n be the restrictionof a to n , that is, a æ n is the element of Ś i ă n α i defined as follows. If a “ p a i q i ă ω , then a æ n “ p a i q i ă n . Here, as usual, we adopt the convention n “ t , , . . . , n ´ u . Let us say that a linear order ă on Ś i ă ω α i is finitely Carruth if ă extends ă ˆ and there are an n ă ω and an order ď extending ă ˆ on Ś i ă n α i and such that if a ‰ b P Ś i ă ω α i and i “ diff p a, b q , then(1) if i ě n , then a ă b if and only if a i ă b i , and(2) if i ă n , then a ă b if and only if a æ n ă a æ n .A linear order ă on Ś i ă ω α i is locally finitely Carruth if ă extends ă ˆ and, for every c P Ś i ă ω α i , there is n “ n c ă ω such that, for every a ‰ b P Ś i ă ω α i , if a, b ă c and i “ diff p a, b q ě n , then a ă b if andonly if a i ă b i . In other words, for every c , (1) above holds, restrictedto those pairs of elements a, b which are ă c , while no version of (2) isassumed. Theorem 3.1. If p α i q i ă ω is a sequence of ordinals, then  i ă ω α i is theorder-type of some finitely Carruth linear order on Ś i ă ω α i .Every (locally) finitely Carruth linear order on Ś i ă ω α i is a well-order; moreover,  i ă ω α i is the largest order-type of all such orderings.Proof. By Corollary 2.10, in particular, equation (6), there is some m ă ω such that  i ă ω α i “ p α b ¨ ¨ ¨ b α m ´ q ¨ ś m ď i ă ω α i . By Car-ruth theorem, there is some linear extension ď C of ď ˆ on Ś i ă m α i such that P “ p Ś i ă m α i , ď C q has order-type  i ă m α i . By the re-sults recalled in Subsection 3.2, ś m ď i ă ω α i is the order-type of P “p Ś m ď i ă ω α i , ď L q . Then P “ L ˚ h ă P h has order-type p α b ¨ ¨ ¨ b α m ´ q ¨ ś m ď i ă ω α i “  i ă ω α i , since, as we mentioned, for two ordinals γ and γ the order-type of L ˚ i ă γ i is γ γ . Through the canonical bi-jection between Ś i ă ω α i and Ś i ă m α i ˆ Ś m ď i ă ω α i , the order ă wehave constructed on L ˚ i ă P i is clearly finitely Carruth. Indeed, in thedefinition of finitely Carruth, take n “ m and take ď as ď C . If i “ diff p a, b q ă m , then the ordering between a and b is determinedby their æ m part, since sequences with the same P -component are or-dered according to their P component. Hence (2) holds. On the otherhand, if i “ diff p a, b q ě m , then a i ă b i if and only if a ă b , by thedefinition of ď L , thus (1) holds. Obviously, ă extends ă ˆ , since both ď C and ď L extend ď ˆ on their respective components. Hence  i ă ω α i can be realized as the order-type of some finitely Carruth order on Ś i ă ω α i and the first statement is proved.Since every finitely Carruth order on Ś i ă ω α i is obviously locallyfinitely Carruth, it is enough to prove that every locally finitely Carruthorder on Ś i ă ω α i is a well-order of type ď  i ă ω α i . So let us assumefrom now on that ă is locally finitely Carruth. N INFINITE NATURAL PRODUCT 13
Claim. If c P Ś i ă ω α i and C “ t a P Ś i ă ω α i | a ă c u , then p C, ă C q is a well-ordered set of type ď Â i ă ω α i .Proof. Let n “ n c be given by local Carruth finiteness and, as above,let P “ p Ś n ď i ă ω α i , ď L q . Notice that, by Subsection 3.2, P is well-ordered and has type ś n ď i ă ω α i . If a P Ś i ă ω α i , say, a “ p a i q i ă ω , recallthat a æ n is the element p a i q i ă n of Ś i ă n α i . Similarly, let a ě n be theelement p a i q i ě n of Ś i ě n α i . Thus the position a ÞÑ p a æ n , a ě n q gives thecanonical bijection (mentioned but not described above) from Ś i ă ω α i to Ś i ă n α i ˆ Ś n ď i ă ω α i . If P “ t d P Ś n ď i ă ω α i | d “ a ě n , for some a P Ś i ă ω α i such that a ă c u , then P Ď P hence P as a suborder of P inherits a well-order of type ď ś n ď i ă ω α i . Moreover, by local Carruthfiniteness, if a, b ă c and a ě n ă L b ě n , then a ă b . If d P P , let Q d “ t a æ n | a P Ś i ă ω α i , a ă c and a ě n “ d u . Then, for every d P P ,the order ă induces an order ă d on Q d by letting a æ n ă d b æ n if andonly if a ă b (notice that, since we are assuming a, b P Q d , then a and b have the same ě n components). Since, by assumption, ă extends ă ˆ on Ś i ă ω α i , then ă d extends the restriction of ă ˆ on Ś i ă n α i to Q d . Hence, by Carruth theorem, for every d P P we have that p Q d , ă d q is well-ordered and has type ď Â i ă n α i .The above considerations show that p C, ă C q is isomorphic to thelexicographical product L d P P Q d (recall that if a, b ă c and a ě n ă L b ě n ,then a ă b ). Since, as we showed, P is a well-ordered set of type ď ś n ď i ă ω α i and each Q d is a well-ordered set of type ď Â i ă n α i , then p C, ă C q is a well-ordered set of order-type ď Â i ă n α i ¨ ś n ď i ă ω α i .If the m given by Corollary 2.10 is ď n , then we immediately getfrom equation (6) that p C, ă C q has order-type ď Â i ă ω α i . Otherwise,notice that if the condition for local Carruth finiteness is satisfied for c and for some n c , then the condition is satisfied for any n ě n c in placeof n c , hence it is no loss of generality to suppose that the m given byCorollary 2.10 is ď n and we are done as before. l Claim
To complete the proof of the theorem, we have from the Claim that,for every c P Ś i ă ω α i , the set of the ă -predecessors of c is well-ordered;this implies that ă is a well-order.It remains to show that ă has order-type ď Â i ă ω α i . This is vac-uously true if some α i is 0 and it follows from Carruth theorem if the α i ’s are eventually 1. Otherwise, local Carruth finiteness implies that ă has no maximum. Indeed, if a P Ś i ă ω α i , then there is k ă ω suchthat a i “
0, for i ą k . Since the sequence of the α i ’s is not eventually 1and no α i is 0, there is ¯ ı ą k such that α ¯ ı ą
1. If b is equal to a on eachcomponent, except that b ¯ ı “
1, then a ă ˆ b , hence a ă b , since, by assumption, ă extends ă ˆ . Since a above has been chosen arbitrarily,we get that ă has no maximum. Then the Claim implies that ă hasorder-type ď Â i ă ω α i . (cid:3) Remark . For the sake of simplicity, we have limited our study hereto sequences of length ω . However, essentially all the results here admita reformulation for the case of ordinal-indexed transfinite sequencesof arbitrary length, modulo the case of the transfinite natural sumsstudied in [L2]. It should be remarked that there are different waysto extend the natural sums and products to sequences of length ą ω .See [L2, Section 5], in particular, Definitions 5.2 and Problems 5.6. Wejust give here the relevant definitions relative to transfinite products.Suppose that p α γ q γ ă ¯ ε is a sequence of ordinals. The iterated naturalproduct ś γ ă δ α γ is defined for every δ ď ¯ ε as follows. ź γ ă α γ “ ź γ ă δ ` α γ “ ˜ ź γ ă δ α γ ¸ b α δ ź γ ă δ α γ “ lim δ ă δ ź γ ă δ α γ for δ limitMoreover, we set â oγ ă δ α γ “ inf π ź γ ă δ α π p γ q where π varies among all the permutations of δ . In the above definitionwe are keeping δ fixed. In the next definition, on the contrary, we letthe ordinal δ vary. Suppose that I is any set and p α i q i P I is a sequenceof ordinals. Define â i P I α i “ inf δ,f ź γ ă δ α f p γ q where δ varies among all the ordinals having cardinality | I | and f variesamong all the bijections from δ to I . Furthermore, let λ “ | I | and define â ‚ i P I α i “ inf f ź γ ă λ α f p γ q where f varies among all the bijections from λ to I . Acknowledgement.
We thank an anonymous referee of [L1] for many in-teresting suggestions concerning the relationship between natural sums
N INFINITE NATURAL PRODUCT 15 and the theory of well-quasi-orders. We thank Harry Altman for stim-ulating discussions. We thank Arnold W. Miller for letting us know ofToulmin paper [T] in the MR review of [L1].The literature on the subject of ordinal operations is so vast andsparse that we cannot claim completeness of the following list of refer-ences.
It is not intended that each work in the list has given equally significant contributions tothe discipline. Henceforth the author disagrees with the use of the list (even in aggregateforms in combination with similar lists) in order to determine rankings or other indicatorsof, e. g., journals, individuals or institutions. In particular, the author considers that itis highly inappropriate, and strongly discourages, the use (even in partial, preliminary orauxiliary forms) of indicators extracted from the list in decisions about individuals (espe-cially, job opportunities, career progressions etc.), attributions of funds, and selections orevaluations of research projects.
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