|
Well Partial Order
In mathematics, specifically order theory, a well-quasi-ordering or wqo is a quasi-ordering such that any infinite sequence of elements x_0, x_1, x_2, \ldots from X contains an increasing pair x_i \leq x_j with i x_2> \cdots) 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 s. Examples 
[...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dickson's Lemma
In mathematics, Dickson's lemma states that every set of n-tuples of natural numbers has finitely many minimal elements. This simple fact from combinatorics has become attributed to the American algebraist L. E. Dickson, who used it to prove a result in number theory about perfect numbers. However, the lemma was certainly known earlier, for example to Paul Gordan in his research on invariant theory.. Example Let K be a fixed number, and let S = \ be the set of pairs of numbers whose product is at least K. When defined over the positive real numbers, S has infinitely many minimal elements of the form (x,K/x), one for each positive number x; this set of points forms one of the branches of a hyperbola. The pairs on this hyperbola are minimal, because it is not possible for a different pair that belongs to S to be less than or equal to (x,K/x) in both of its coordinates. However, Dickson's lemma concerns only tuples of natural numbers, and over the natural numbers there are only fini ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Laver's Theorem
Laver's theorem, in order theory, states that order embeddability of countable total orders is a well-quasi-ordering. That is, for every infinite sequence of totally-ordered countable sets, there exists an order embedding from an earlier member of the sequence to a later member. This result was previously known as Fraïssé's conjecture, after Roland Fraïssé, who conjectured it in 1948; Richard Laver proved the conjecture in 1971. More generally, Laver proved the same result for order embeddings of countable unions of scattered orders. In reverse mathematics, the version of the theorem for countable orders is denoted FRA (for Fraïssé) and the version for countable unions of scattered orders is denoted LAV (for Laver). In terms of the "big five" systems of second-order arithmetic, FRA is known to fall in strength somewhere between the strongest two systems, \Pi_1^1-CA0 and ATR0, and to be weaker than \Pi_1^1-CA0. However, it remains open whether it is equivalent to ATR0 or stric ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Linear Order
In mathematics, a total or linear order is a partial order in which any two elements are comparable. That is, a total order is a binary relation \leq on some set X, which satisfies the following for all a, b and c in X: # a \leq a ( reflexive). # If a \leq b and b \leq c then a \leq c ( transitive). # If a \leq b and b \leq a then a = b ( antisymmetric). # a \leq b or b \leq a (strongly connected, formerly called total). Total orders are sometimes also called simple, connex, or full orders. A set equipped with a total order is a totally ordered set; the terms simply ordered set, linearly ordered set, and loset are also used. The term ''chain'' is sometimes defined as a synonym of ''totally ordered set'', but refers generally to some sort of totally ordered subsets of a given partially ordered set. An extension of a given partial order to a total order is called a linear extension of that partial order. Strict and non-strict total orders A on a set X is a strict partial or ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Scattered Order
In mathematical order theory, a scattered order is a linear order that contains no densely ordered subset with more than one element. A characterization due to Hausdorff states that the class of all scattered orders is the smallest class of linear orders that contains the singleton orders and is closed under well-ordered and reverse well-ordered sums. Laver's theorem (generalizing a conjecture of Roland Fraïssé on countable orders) states that the embedding relation on the class of countable unions of scattered orders is a well-quasi-order.Harzheim, Theorem 6.17, p. 201; The order topology of a scattered order is scattered. The converse implication does not hold, as witnessed by the lexicographic order In mathematics, the lexicographic or lexicographical order (also known as lexical order, or dictionary order) is a generalization of the alphabetical order of the dictionaries to sequences of ordered symbols or, more generally, of elements of a ... on \mathbb Q\times\ma ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Crispin St
Saints Crispin and Crispinian are the Christian patron saints of cobblers, curriers, tanners, and leather workers. They were beheaded during the reign of Diocletian; the date of their execution is given as 25 October 285 or 286. History Born to a noble Roman family in the 3rd century AD, Crispin and Crispinian fled persecution for their faith, ending up at Soissons, where they preached Christianity to the Gauls while making shoes by night. It is stated that they were twin brothers. They earned enough by their trade to support themselves and also to aid the poor. Their success attracted the ire of Rictus Varus, governor of Belgic Gaul, who had them tortured and thrown into the river with millstones around their necks. Though they survived, they were beheaded by the Emperor 286. Veneration The feast day of Saints Crispin and Crispinian is 25 October. Although this feast was removed from the Roman Catholic Church's universal liturgical calendar following the Second Vatican Coun ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kruskal's Tree Theorem
In mathematics, Kruskal's tree theorem states that the set of finite trees over a well-quasi-ordered set of labels is itself well-quasi-ordered under homeomorphic embedding. History The theorem was conjectured by Andrew Vázsonyi and proved by ; a short proof was given by . It has since become a prominent example in reverse mathematics as a statement that cannot be proved within ATR0 (a form of arithmetical transfinite recursion), and a finitary application of the theorem gives the existence of the fast-growing TREE function. In 2004, the result was generalized from trees to graphs as the Robertson–Seymour theorem, a result that has also proved important in reverse mathematics and leads to the even-faster-growing SSCG function. Statement The version given here is that proven by Nash-Williams; Kruskal's formulation is somewhat stronger. All trees we consider are finite. Given a tree with a root, and given vertices , , call a successor of if the unique path from the root ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Better-quasi-ordering
In order theory a better-quasi-ordering or bqo is a quasi-ordering that does not admit a certain type of bad array. Every better-quasi-ordering is a well-quasi-ordering. Motivation Though ''well-quasi-ordering'' is an appealing notion, many important infinitary operations do not preserve well-quasi-orderedness. An example due to Richard Rado illustrates this. In a 1965 paper Crispin Nash-Williams formulated the stronger notion of ''better-quasi-ordering'' in order to prove that the class of trees of height ω is well-quasi-ordered under the ''topological minor'' relation. Since then, many quasi-orderings have been proven to be well-quasi-orderings by proving them to be better-quasi-orderings. For instance, Richard Laver established Laver's theorem (previously a conjecture of Roland Fraïssé) by proving that the class of scattered linear order types is better-quasi-ordered. More recently, Carlos Martinez-Ranero has proven that, under the proper forcing axiom, the class of Aron ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Higman's Lemma
In mathematics, Higman's lemma states that the set of finite sequences over a finite alphabet, as partially ordered by the subsequence relation, is well-quasi-ordered. That is, if w_1, w_2, \ldots is an infinite sequence of words over some fixed finite alphabet, then there exist indices i < j such that can be obtained from by deleting some (possibly none) symbols. More generally this remains true when the alphabet is not necessarily finite, but is itself well-quasi-ordered, and the subsequence relation allows the replacement of symbols by earlier symbols in the well-quasi-ordering of labels. This is a special case of the later . It is named after Graham Higman
[...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Embedding
In mathematics, an embedding (or imbedding) is one instance of some mathematical structure contained within another instance, such as a group that is a subgroup. When some object X is said to be embedded in another object Y, the embedding is given by some injective and structure-preserving map f:X\rightarrow Y. The precise meaning of "structure-preserving" depends on the kind of mathematical structure of which X and Y are instances. In the terminology of category theory, a structure-preserving map is called a morphism. The fact that a map f:X\rightarrow Y is an embedding is often indicated by the use of a "hooked arrow" (); thus: f : X \hookrightarrow Y. (On the other hand, this notation is sometimes reserved for inclusion maps.) Given X and Y, several different embeddings of X in Y may be possible. In many cases of interest there is a standard (or "canonical") embedding, like those of the natural numbers in the integers, the integers in the rational numbers, the rational n ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Subsequence
In mathematics, a subsequence of a given sequence is a sequence that can be derived from the given sequence by deleting some or no elements without changing the order of the remaining elements. For example, the sequence \langle A,B,D \rangle is a subsequence of \langle A,B,C,D,E,F \rangle obtained after removal of elements C, E, and F. The relation of one sequence being the subsequence of another is a preorder. Subsequences can contain consecutive elements which were not consecutive in the original sequence. A subsequence which consists of a consecutive run of elements from the original sequence, such as \langle B,C,D \rangle, from \langle A,B,C,D,E,F \rangle, is a substring. The substring is a refinement of the subsequence. The list of all subsequences for the word "apple" would be "''a''", "''ap''", "''al''", "''ae''", "''app''", "''apl''", "''ape''", "''ale''", "''appl''", "''appe''", "''aple''", "''apple''", "''p''", "''pp''", "''pl''", "''pe''", "''ppl''", "''ppe''", "''ple' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Prefix (computer Science)
In formal language theory and computer science, a substring is a contiguous sequence of characters within a string. For instance, "''the best of''" is a substring of "''It was the best of times''". In contrast, "''Itwastimes''" is a subsequence of "''It was the best of times''", but not a substring. Prefixes and suffixes are special cases of substrings. A prefix of a string S is a substring of S that occurs at the beginning of S; likewise, a suffix of a string S is a substring that occurs at the end of S. The substrings of the string "''apple''" would be: "''a''", "''ap''", "''app''", "''appl''", "''apple''", "''p''", "''pp''", "''ppl''", "''pple''", "''pl''", "''ple''", "''l''", "''le''" "''e''", "" (note the empty string at the end). Substring A string u is a substring (or factor) of a string t if there exists two strings p and s such that t = pus. In particular, the empty string is a substring of every string. Example: The string u=ana is equal to substrings (and subse ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
