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Well-partial-order
In mathematics, specifically order theory, a well-quasi-ordering or wqo on a set X is a quasi-ordering of X for which every 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 and has no infinite s. Ordinal type 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' ...[...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...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 natural 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 on ... [...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 In mathematical logic, second-order arithmetic is a collection of axiomatic systems that formalize the natural numbers and their subsets. It is an alternative to axiomatic set theory as a foundation ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Linear Order
In mathematics, a total order 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 totality). Requirements 1. to 3. just make up the definition of a partial order. Reflexivity (1.) already follows from strong connectedness (4.), but is required explicitly by many authors nevertheless, to indicate the kinship to partial orders. 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, toset and loset are also used. The term ''chain'' is sometimes defined as a synonym of ''totally ordered set'', but generally refers to a ... [...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. Legend 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 aid people experiencing poverty. 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 S ... [...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. A finitary application of the theorem gives the existence of the fast-growing TREE function. TREE(3) is largely accepted to be one of the largest simply defined finite numbers, dwarfing other large numbers such as Graham's number and googolplex. 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 in ATR0 (a second-order arithmetic theory with a form of arithmetical transfinite recursion). 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, which dwarfs \text(3). Statement The version given here is that p ... [...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 Tree (set theory), trees of height Ordinal number#Ordinals extend the natural numbers, 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-Rane ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Higman's Lemma
In mathematics, Higman's lemma states that the set \Sigma^* of finite sequences over a finite alphabet \Sigma, as partially ordered by the subsequence relation, is a well partial order. That is, if w_1, w_2, \ldots\in \Sigma^* is an infinite sequence of words over a finite alphabet \Sigma, then there exist indices i < j such that can be obtained from by deleting some (possibly none) symbols. More generally the set of sequences is well-quasi-ordered even when is not necessarily finite, but is itself well-quasi-ordered, and the subsequence ordering is generalized into an "embedding" quasi-order that allows the replacement of symbols by earlier symbols in the well-quasi-ordering of . This is a special case of the later . It is named after |
Embedding
In mathematics, an embedding (or imbedding) is one instance of some mathematical structure contained within another instance, such as a group (mathematics), 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 function, 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 i ... [...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 partial order. 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''", " ... [...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 sub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
