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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] |
Order Theory
Order theory is a branch of mathematics that investigates the intuitive notion of order using binary relations. It provides a formal framework for describing statements such as "this is less than that" or "this precedes that". This article introduces the field and provides basic definitions. A list of order-theoretic terms can be found in the order theory glossary. Background and motivation Orders are everywhere in mathematics and related fields like computer science. The first order often discussed in primary school is the standard order on the natural numbers e.g. "2 is less than 3", "10 is greater than 5", or "Does Tom have fewer cookies than Sally?". This intuitive concept can be extended to orders on other sets of numbers, such as the integers and the reals. The idea of being greater than or less than another number is one of the basic intuitions of number systems (compare with numeral systems) in general (although one usually is also interested in the actual difference ... [...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] |
Dense Order
In mathematics, a partial order or total order < on a is said to be dense if, for all and in for which , there is a in such that . That is, for any two elements, one less than the other, there is another element between them. For total orders this can be simplified to "for any two distinct elements, there is another element between them", since all elements of a total order are . Example The s as a linearly ordered set are a densely o ...[...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Subset
In mathematics, Set (mathematics), set ''A'' is a subset of a set ''B'' if all Element (mathematics), elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are unequal, then ''A'' is a proper subset of ''B''. The relationship of one set being a subset of another is called inclusion (or sometimes containment). ''A'' is a subset of ''B'' may also be expressed as ''B'' includes (or contains) ''A'' or ''A'' is included (or contained) in ''B''. A ''k''-subset is a subset with ''k'' elements. The subset relation defines a partial order on sets. In fact, the subsets of a given set form a Boolean algebra (structure), Boolean algebra under the subset relation, in which the join and meet are given by Intersection (set theory), intersection and Union (set theory), union, and the subset relation itself is the Inclusion (Boolean algebra), Boolean inclusion relation. Definition If ''A'' and ''B'' are sets and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Felix Hausdorff
Felix Hausdorff ( , ; November 8, 1868 – January 26, 1942) was a German mathematician who is considered to be one of the founders of modern topology and who contributed significantly to set theory, descriptive set theory, measure theory, and functional analysis. Life became difficult for Hausdorff and his family after Kristallnacht in 1938. The next year he initiated efforts to emigrate to the United States, but was unable to make arrangements to receive a research fellowship. On 26 January 1942, Felix Hausdorff, along with his wife and his sister-in-law, died by suicide by taking an overdose of veronal, rather than comply with German orders to move to the Endenich camp, and there suffer the likely implications, about which he held no illusions. Life Childhood and youth Hausdorff's father, the Jewish merchant Louis Hausdorff (1843–1896), moved with his young family to Leipzig in the autumn of 1870, and over time worked at various companies, including a linen-and cotton goo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Well-order
In mathematics, a well-order (or well-ordering or well-order relation) on a set ''S'' is a total order on ''S'' with the property that every non-empty subset of ''S'' has a least element in this ordering. The set ''S'' together with the well-order relation is then called a well-ordered set. In some academic articles and textbooks these terms are instead written as wellorder, wellordered, and wellordering or well order, well ordered, and well ordering. Every non-empty well-ordered set has a least element. Every element ''s'' of a well-ordered set, except a possible greatest element, has a unique successor (next element), namely the least element of the subset of all elements greater than ''s''. There may be elements besides the least element which have no predecessor (see below for an example). A well-ordered set ''S'' contains for every subset ''T'' with an upper bound a least upper bound, namely the least element of the subset of all upper bounds of ''T'' in ''S''. If ≤ is a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Total 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 ord ... [...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] |
Roland Fraïssé
Roland Fraïssé (; 12 March 1920 – 30 March 2008) was a French mathematical logician. Fraïssé received his doctoral degree from the University of Paris in 1953. In his thesis, Fraïssé used the back-and-forth method to determine whether two model-theoretic structures were elementarily equivalent. This method of determining elementary equivalence was later formulated as the Ehrenfeucht–Fraïssé game. Fraïssé worked primarily in relation theory. Another of his important works was the Fraïssé construction of a Fraïssé limit of finite structures. He also formulated Fraïssé's conjecture on order embeddings, and introduced the notion of compensor in the theory of posets.Petits posets : dénombrement, représentabilité par cercles et compenseurs, Roland Fraïssé and Nik Lygeros, ''Comptes Rendus de l'Académie des Sciences'', Série I 313 (1991), no. 7, 417–420 Most of his career was spent as Professor at the University of Provence in Marseille Marseille ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Countable
In mathematics, a set is countable if either it is finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is ''countable'' if there exists an injective function from it into the natural numbers; this means that each element in the set may be associated to a unique natural number, or that the elements of the set can be counted one at a time, although the counting may never finish due to an infinite number of elements. In more technical terms, assuming the axiom of countable choice, a set is ''countable'' if its cardinality (its number of elements) is not greater than that of the natural numbers. A countable set that is not finite is said countably infinite. The concept is attributed to Georg Cantor, who proved the existence of uncountable sets, that is, sets that are not countable; for example the set of the real numbers. A note on terminology Although the terms "countable" and "countably infinite" as defined here are quite comm ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Well-quasi-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 and has no infinite s. Examples 
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Annals Of Mathematics
The ''Annals of Mathematics'' is a mathematical journal published every two months by Princeton University and the Institute for Advanced Study. History The journal was established as ''The Analyst'' in 1874 and with Joel E. Hendricks as the founding editor-in-chief. It was "intended to afford a medium for the presentation and analysis of any and all questions of interest or importance in pure and applied Mathematics, embracing especially all new and interesting discoveries in theoretical and practical astronomy, mechanical philosophy, and engineering". It was published in Des Moines, Iowa, and was the earliest American mathematics journal to be published continuously for more than a year or two. This incarnation of the journal ceased publication after its tenth year, in 1883, giving as an explanation Hendricks' declining health, but Hendricks made arrangements to have it taken over by new management, and it was continued from March 1884 as the ''Annals of Mathematics''. The n ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
