Transitive binary relations
Symmetric Antisymmetric Connected Well-founded Has joins Has meets Reflexive Irreflexive Asymmetric
Total, Semiconnex Anti-
Equivalence relation Green tickY Green tickY
Preorder (Quasiorder) Green tickY
Partial order Green tickY Green tickY
Total preorder Green tickY Green tickY
Total order Green tickY Green tickY Green tickY
Prewellordering Green tickY Green tickY Green tickY
Well-quasi-ordering Green tickY Green tickY
Well-ordering Green tickY Green tickY Green tickY Green tickY
Lattice Green tickY Green tickY Green tickY Green tickY
Join-semilattice Green tickY Green tickY Green tickY
Meet-semilattice Green tickY Green tickY Green tickY
Strict partial order Green tickY Green tickY Green tickY
Strict weak order Green tickY Green tickY Green tickY
Strict total order Green tickY Green tickY Green tickY Green tickY
Symmetric Antisymmetric Connected Well-founded Has joins Has meets Reflexive Irreflexive Asymmetric
Definitions, for all and
Green tickY indicates that the column's property is always true the row's term (at the very left), while indicates that the property is not guaranteed in general (it might, or might not, hold). For example, that every equivalence relation is symmetric, but not necessarily antisymmetric, is indicated by Green tickY in the "Symmetric" column and in the "Antisymmetric" column, respectively.

All definitions tacitly require the homogeneous relation be transitive: for all if and then
A term's definition may require additional properties that are not listed in this table.

In mathematics, specifically order theory, a well-quasi-ordering or wqo on a set is a quasi-ordering of for which every infinite sequence of elements from contains an increasing pair with

Motivation edit

Well-founded induction can be used on any set with a well-founded relation, thus one is interested in when a quasi-order is well-founded. (Here, by abuse of terminology, a quasiorder   is said to be well-founded if the corresponding strict order   is a well-founded relation.) However the class of well-founded quasiorders is not closed under certain operations—that is, when a quasi-order is used to obtain a new quasi-order on a set of structures derived from our original set, this quasiorder is found to be not well-founded. By placing stronger restrictions on the original well-founded quasiordering one can hope to ensure that our derived quasiorderings are still well-founded.

An example of this is the power set operation. Given a quasiordering   for a set   one can define a quasiorder   on  's power set   by setting   if and only if for each element of   one can find some element of   that is larger than it with respect to  . One can show that this quasiordering on   needn't be well-founded, but if one takes the original quasi-ordering to be a well-quasi-ordering, then it is.

Formal definition edit

A well-quasi-ordering on a set   is a quasi-ordering (i.e., a reflexive, transitive binary relation) such that any infinite sequence of elements   from   contains an increasing pair   with  . The set   is said to be well-quasi-ordered, or shortly wqo.

A well partial order, or a wpo, is a wqo that is a proper ordering relation, i.e., it is antisymmetric.

Among other ways of defining wqo's, one is to say that they are quasi-orderings which do not contain infinite strictly decreasing sequences (of the form  )[1] nor infinite sequences of pairwise incomparable elements. Hence a quasi-order (X, ≤) is wqo if and only if (X, <) is well-founded and has no infinite antichains.

Ordinal type edit

Let   be well partially ordered. A (necessarily finite) sequence   of elements of   that contains no pair   with   is usually called a bad sequence. The tree of bad sequences   is the tree that contains a vertex for each bad sequence, and an edge joining each nonempty bad sequence   to its parent  . The root of   corresponds to the empty sequence. Since   contains no infinite bad sequence, the tree   contains no infinite path starting at the root.[citation needed] Therefore, each vertex   of   has an ordinal height  , which is defined by transfinite induction as  . The ordinal type of  , denoted  , is the ordinal height of the root of  .

A linearization of   is an extension of the partial order into a total order. It is easy to verify that   is an upper bound on the ordinal type of every linearization of  . De Jongh and Parikh[1] proved that in fact there always exists a linearization of   that achieves the maximal ordinal type  .

Examples edit

Pic.1: Integer numbers with the usual order
Pic.2: Hasse diagram of the natural numbers ordered by divisibility
Pic.3: Hasse diagram of   with componentwise order
  •  , the set of natural numbers with standard ordering, is a well partial order (in fact, a well-order). However,  , the set of positive and negative integers, is not a well-quasi-order, because it is not well-founded (see Pic.1).
  •  , the set of natural numbers ordered by divisibility, is not a well-quasi-order: the prime numbers are an infinite antichain (see Pic.2).
  •  , the set of vectors of   natural numbers (where   is finite) with component-wise ordering, is a well partial order (Dickson's lemma; see Pic.3). More generally, if   is well-quasi-order, then   is also a well-quasi-order for all  .
  • Let   be an arbitrary finite set with at least two elements. The set   of words over   ordered lexicographically (as in a dictionary) is not a well-quasi-order because it contains the infinite decreasing sequence  . Similarly,   ordered by the prefix relation is not a well-quasi-order, because the previous sequence is an infinite antichain of this partial order. However,   ordered by the subsequence relation is a well partial order.[2] (If   has only one element, these three partial orders are identical.)
  • More generally,  , the set of finite  -sequences ordered by embedding is a well-quasi-order if and only if   is a well-quasi-order (Higman's lemma). Recall that one embeds a sequence   into a sequence   by finding a subsequence of   that has the same length as   and that dominates it term by term. When   is an unordered set,   if and only if   is a subsequence of  .
  •  , the set of infinite sequences over a well-quasi-order  , ordered by embedding, is not a well-quasi-order in general. That is, Higman's lemma does not carry over to infinite sequences. Better-quasi-orderings have been introduced to generalize Higman's lemma to sequences of arbitrary lengths.
  • Embedding between finite trees with nodes labeled by elements of a wqo   is a wqo (Kruskal's tree theorem).
  • Embedding between infinite trees with nodes labeled by elements of a wqo   is a wqo (Nash-Williams' theorem).
  • Embedding between countable scattered linear order types is a well-quasi-order (Laver's theorem).
  • Embedding between countable boolean algebras is a well-quasi-order. This follows from Laver's theorem and a theorem of Ketonen.
  • Finite graphs ordered by a notion of embedding called "graph minor" is a well-quasi-order (Robertson–Seymour theorem).
  • Graphs of finite tree-depth ordered by the induced subgraph relation form a well-quasi-order,[3] as do the cographs ordered by induced subgraphs.[4]

Constructing new wpo's from given ones edit

Let   and   be two disjoint wpo sets. Let  , and define a partial order on   by letting   if and only if   for the same   and  . Then   is wpo, and  , where   denotes natural sum of ordinals.[1]

Given wpo sets   and  , define a partial order on the Cartesian product  , by letting   if and only if   and  . Then   is wpo (this is a generalization of Dickson's lemma), and  , where   denotes natural product of ordinals.[1]

Given a wpo set  , let   be the set of finite sequences of elements of  , partially ordered by the subsequence relation. Meaning, let   if and only if there exist indices   such that   for each  . By Higman's lemma,   is wpo. The ordinal type of   is[1][5]


Given a wpo set  , let   be the set of all finite rooted trees whose vertices are labeled by elements of  . Partially order   by the tree embedding relation. By Kruskal's tree theorem,   is wpo. This result is nontrivial even for the case   (which corresponds to unlabeled trees), in which case   equals the small Veblen ordinal. In general, for   countable, we have the upper bound   in terms of the   ordinal collapsing function. (The small Veblen ordinal equals   in this ordinal notation.)[6]

Wqo's versus well partial orders edit

In practice, the wqo's one manipulates are quite often not orderings (see examples above), and the theory is technically smoother[citation needed] if we do not require antisymmetry, so it is built with wqo's as the basic notion. On the other hand, according to Milner 1985, no real gain in generality is obtained by considering quasi-orders rather than partial orders... it is simply more convenient to do so.

Observe that a wpo is a wqo, and that a wqo gives rise to a wpo between equivalence classes induced by the kernel of the wqo. For example, if we order   by divisibility, we end up with   if and only if  , so that  .

Infinite increasing subsequences edit

If   is wqo then every infinite sequence   contains an infinite increasing subsequence   (with  ). Such a subsequence is sometimes called perfect. This can be proved by a Ramsey argument: given some sequence  , consider the set   of indexes   such that   has no larger or equal   to its right, i.e., with  . If   is infinite, then the  -extracted subsequence contradicts the assumption that   is wqo. So   is finite, and any   with   larger than any index in   can be used as the starting point of an infinite increasing subsequence.

The existence of such infinite increasing subsequences is sometimes taken as a definition for well-quasi-ordering, leading to an equivalent notion.

Properties of wqos edit

  • Given a quasiordering   the quasiordering   defined by   is well-founded if and only if   is a wqo.[7]
  • A quasiordering is a wqo if and only if the corresponding partial order (obtained by quotienting by  ) has no infinite descending sequences or antichains. (This can be proved using a Ramsey argument as above.)
  • Given a well-quasi-ordering  , any sequence of upward-closed subsets   eventually stabilises (meaning there exists   such that  ; a subset   is called upward-closed if  ): assuming the contrary  , a contradiction is reached by extracting an infinite non-ascending subsequence.
  • Given a well-quasi-ordering  , any subset   of   has a finite number of minimal elements with respect to  , for otherwise the minimal elements of   would constitute an infinite antichain.

See also edit

Notes edit

^ Here x < y means:   and  

References edit

  1. ^ a b c d de Jongh, Dick H. G.; Parikh, Rohit (1977). "Well-partial orderings and hierarchies". Indagationes Mathematicae (Proceedings). 80 (3): 195–207. doi:10.1016/1385-7258(77)90067-1.
  2. ^ Gasarch, W. (1998), "A survey of recursive combinatorics", Handbook of Recursive Mathematics, Vol. 2, Stud. Logic Found. Math., vol. 139, Amsterdam: North-Holland, pp. 1041–1176, doi:10.1016/S0049-237X(98)80049-9, MR 1673598. See in particular page 1160.
  3. ^ Nešetřil, Jaroslav; Ossona de Mendez, Patrice (2012), "Lemma 6.13", Sparsity: Graphs, Structures, and Algorithms, Algorithms and Combinatorics, vol. 28, Heidelberg: Springer, p. 137, doi:10.1007/978-3-642-27875-4, ISBN 978-3-642-27874-7, MR 2920058.
  4. ^ Damaschke, Peter (1990), "Induced subgraphs and well-quasi-ordering", Journal of Graph Theory, 14 (4): 427–435, doi:10.1002/jgt.3190140406, MR 1067237.
  5. ^ Schmidt, Diana (1979). Well-partial orderings and their maximal order types (Habilitationsschrift). Heidelberg. Republished in: Schmidt, Diana (2020). "Well-partial orderings and their maximal order types". In Schuster, Peter M.; Seisenberger, Monika; Weiermann, Andreas (eds.). Well-Quasi Orders in Computation, Logic, Language and Reasoning. Trends in Logic. Vol. 53. Springer. pp. 351–391. doi:10.1007/978-3-030-30229-0_13.
  6. ^ Rathjen, Michael; Weiermann, Andreas (1993). "Proof-theoretic investigations on Kruskal's theorem". Annals of Pure and Applied Logic. 60: 49–88. doi:10.1016/0168-0072(93)90192-G.
  7. ^ Forster, Thomas (2003). "Better-quasi-orderings and coinduction". Theoretical Computer Science. 309 (1–3): 111–123. doi:10.1016/S0304-3975(03)00131-2.