Critical Pair (order Theory)

picture info	Critical Pair (order Theory) In order theory, a discipline within mathematics, a critical pair is a pair of elements in a partially ordered set that are incomparable but that could be made comparable without requiring any other changes to the partial order. Formally, let be a partially ordered set. Then a critical pair is an ordered pair of elements of with the following three properties: * and are incomparable in , for every in , if then , and for every in , if then . If is a critical pair, then the binary relation obtained from by adding the single relationship is also a partial order. The properties required of critical pairs ensure that, when the relationship is added, the addition does not cause any violations of the transitive property. A set of linear extension In order theory, a branch of mathematics, a linear extension of a partial order is a total order (or linear order) that is compatible with the partial order. As a classic example, the lexicographic order of totally ordered set ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
picture info	Critical Pair (order Theory) In order theory, a discipline within mathematics, a critical pair is a pair of elements in a partially ordered set that are incomparable but that could be made comparable without requiring any other changes to the partial order. Formally, let be a partially ordered set. Then a critical pair is an ordered pair of elements of with the following three properties: * and are incomparable in , for every in , if then , and for every in , if then . If is a critical pair, then the binary relation obtained from by adding the single relationship is also a partial order. The properties required of critical pairs ensure that, when the relationship is added, the addition does not cause any violations of the transitive property. A set of linear extension In order theory, a branch of mathematics, a linear extension of a partial order is a total order (or linear order) that is compatible with the partial order. As a classic example, the lexicographic order of totally ordered set ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
picture info	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]
picture info	Partially Ordered Set In mathematics, especially order theory, a partially ordered set (also poset) formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a Set (mathematics), set. A poset consists of a set together with a binary relation indicating that, for certain pairs of elements in the set, one of the elements precedes the other in the ordering. The relation itself is called a "partial order." The word ''partial'' in the names "partial order" and "partially ordered set" is used as an indication that not every pair of elements needs to be comparable. That is, there may be pairs of elements for which neither element precedes the other in the poset. Partial orders thus generalize total orders, in which every pair is comparable. Informal definition A partial order defines a notion of Comparability, comparison. Two elements ''x'' and ''y'' may stand in any of four mutually exclusive relationships to each other: either ''x'' ''y'', ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
picture info	Comparability In mathematics, two elements ''x'' and ''y'' of a set ''P'' are said to be comparable with respect to a binary relation ≤ if at least one of ''x'' ≤ ''y'' or ''y'' ≤ ''x'' is true. They are called incomparable if they are not comparable. Rigorous definition A binary relation on a set P is by definition any subset R of P \times P. Given x, y \in P, x R y is written if and only if (x, y) \in R, in which case x is said to be to y by R. An element x \in P is said to be , or (), to an element y \in P if x R y or y R x. Often, a symbol indicating comparison, such as \,,\, \geq, and many others) is used instead of R, in which case x < y is written in place of $x R y,$ which is why the term "comparable" is used. Comparability with respect to $R$ induces a canonical binary relation on $P$ ; specifically, the induced by $R$ is defined to be the set of all pairs $(x, y) \in P \times P$ such that $x$ i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
picture info	Transitive Property In mathematics, a relation on a set is transitive if, for all elements , , in , whenever relates to and to , then also relates to . Each partial order as well as each equivalence relation needs to be transitive. Definition A homogeneous relation on the set is a ''transitive relation'' if, :for all , if and , then . Or in terms of first-order logic: :\forall a,b,c \in X: (aRb \wedge bRc) \Rightarrow aRc, where is the infix notation for . Examples As a non-mathematical example, the relation "is an ancestor of" is transitive. For example, if Amy is an ancestor of Becky, and Becky is an ancestor of Carrie, then Amy, too, is an ancestor of Carrie. On the other hand, "is the birth parent of" is not a transitive relation, because if Alice is the birth parent of Brenda, and Brenda is the birth parent of Claire, then this does not imply that Alice is the birth parent of Claire. What is more, it is antitransitive: Alice can ''never'' be the birth parent of Claire. "Is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Linear Extension In order theory, a branch of mathematics, a linear extension of a partial order is a total order (or linear order) that is compatible with the partial order. As a classic example, the lexicographic order of totally ordered sets is a linear extension of their product order. Definitions Given any partial orders \,\leq\, and \,\leq^\, on a set X, \,\leq^\, is a linear extension of \,\leq\, exactly when (1) \,\leq^\, is a total order and (2) for every x, y \in X, if x \leq y, then x \leq^ y. It is that second property that leads mathematicians to describe \,\leq^\, as extending \,\leq. Alternatively, a linear extension may be viewed as an order-preserving bijection from a partially ordered set P to a chain C on the same ground set. Order-extension principle The statement that every partial order can be extended to a total order is known as the order-extension principle. A proof using the axiom of choice was first published by Edward Marczewski in 1930. Marczewski write ... [...More Info...] [...Related Items...] OR:* [Wikipedia] [Google] [Baidu]
	Order Dimension In mathematics, the dimension of a partially ordered set (poset) is the smallest number of total orders the intersection of which gives rise to the partial order. This concept is also sometimes called the order dimension or the Dushnik–Miller dimension of the partial order. first studied order dimension; for a more detailed treatment of this subject than provided here, see . Formal definition The dimension of a poset ''P'' is the least integer ''t'' for which there exists a family :\mathcal R=(<_1,\dots,<_t) of linear extension s of ''P'' so that, for every ''x'' and ''y'' in ''P'', ''x'' precedes ''y'' in ''P'' if and only if it precedes ''y'' in all of the linear extensions. That is, : $P=\bigcap\mathcal R=\bigcap_^t <_i.$ An alternative definition of order dimension is the minimal number of [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]