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Morley Rank
In mathematical logic, Morley rank, introduced by , is a means of measuring the size of a subset of a model of a theory, generalizing the notion of dimension in algebraic geometry. Definition Fix a theory ''T'' with a model ''M''. The Morley rank of a formula ''φ'' defining a definable (with parameters) subset ''S'' of ''M'' is an ordinal or −1 or ∞, defined by first recursively defining what it means for a formula to have Morley rank at least ''α'' for some ordinal ''α''. *The Morley rank is at least 0 if ''S'' is non-empty. *For ''α'' a successor ordinal, the Morley rank is at least ''α'' if in some elementary extension ''N'' of ''M'', the set ''S'' has countably infinitely many disjoint definable subsets ''Si'', each of rank at least ''α'' − 1. *For ''α'' a non-zero limit ordinal, the Morley rank is at least ''α'' if it is at least ''β'' for all ''β'' less than ''α''. The Morley rank is then defined to be ''α'' if it is at least ''α'' but ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical Logic
Mathematical logic is the study of logic, formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory. Research in mathematical logic commonly addresses the mathematical properties of formal systems of logic such as their expressive or deductive power. However, it can also include uses of logic to characterize correct mathematical reasoning or to establish foundations of mathematics. Since its inception, mathematical logic has both contributed to and been motivated by the study of foundations of mathematics. This study began in the late 19th century with the development of axiomatic frameworks for geometry, arithmetic, and Mathematical analysis, analysis. In the early 20th century it was shaped by David Hilbert's Hilbert's program, program to prove the consistency of foundational theories. Results of Kurt Gödel, Gerhard Gentzen, and others provided partial resolution to the program, and clarified the issues involved in pr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebraically Closed Field
In mathematics, a field is algebraically closed if every non-constant polynomial in (the univariate polynomial ring with coefficients in ) has a root in . Examples As an example, the field of real numbers is not algebraically closed, because the polynomial equation ''x''2 + 1 = 0 has no solution in real numbers, even though all its coefficients (1 and 0) are real. The same argument proves that no subfield of the real field is algebraically closed; in particular, the field of rational numbers is not algebraically closed. Also, no finite field ''F'' is algebraically closed, because if ''a''1, ''a''2, ..., ''an'' are the elements of ''F'', then the polynomial (''x'' − ''a''1)(''x'' − ''a''2) ⋯ (''x'' − ''a''''n'') + 1 has no zero in ''F''. By contrast, the fundamental theorem of algebra states that the field of complex numbers is algebraically closed. Another example of an algebraicall ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ali Nesin
ʿAlī ibn Abī Ṭālib ( ar, عَلِيّ بْن أَبِي طَالِب; 600 – 661 CE) was the last of four Rightly Guided Caliphs to rule Islam (r. 656 – 661) immediately after the death of Muhammad, and he was the first Shia Imam. The issue of his succession caused a major rift between Muslims and divided them into Shia and Sunni groups. Ali was assassinated in the Grand Mosque of Kufa in 661 by the forces of Mu'awiya, who went on to found the Umayyad Caliphate. The Imam Ali Shrine and the city of Najaf were built around Ali's tomb and it is visited yearly by millions of devotees. Ali was a cousin and son-in-law of Muhammad, raised by him from the age of 5, and accepted his claim of divine revelation by age 11, being among the first to do so. Ali played a pivotal role in the early years of Islam while Muhammad was in Mecca and under severe persecution. After Muhammad's relocation to Medina in 622, Ali married his daughter Fatima and, among others, fathered Hasan ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Alexandre Borovik
Alexandre V. Borovik (born 1956) is a Professor of Pure Mathematics at the University of Manchester, United Kingdom. He was born in Russia and graduated from Novosibirsk State University in 1978. His principal research lies in algebra Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics. Elementary ..., model theory, and combinatorics—topics on which he published several monographs and a number of papers.See Borovik et al (2011) He also has an interest in mathematical practice: his book ''Mathematics under the Microscope: Notes on Cognitive Aspects of Mathematical Practice'' examines a mathematician's outlook on psychophysiological and cognitive issues in mathematics. Selected books and articles * *. * Borovik, Alexandre; Nesin, Ali: ''Groups of finite Morley rank''. Oxford Logic Guides, 26. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
U-rank
In model theory, a branch of mathematical logic, U-rank is one measure of the complexity of a (complete) type, in the context of stable theories. As usual, higher U-rank indicates less restriction, and the existence of a U-rank for all types over all sets is equivalent to an important model-theoretic condition: in this case, superstability. Definition U-rank is defined inductively, as follows, for any (complete) n-type p over any set A: * ''U''(''p'') ≥ 0 * If ''δ'' is a limit ordinal, then ''U''(''p'') ≥ ''δ'' precisely when ''U''(''p'') ≥ ''α'' for all ''α'' less than ''δ'' * For any ''α'' = ''β'' + 1, ''U''(''p'') ≥ ''α'' precisely when there is a forking extension ''q'' of ''p'' with ''U''(''q'') ≥ ''β'' We say that ''U''(''p'') = ''α'' when the ''U''(''p'') ≥ ''α'' but not ''U''(''p'') ≥ ''α'' + 1. If ''U''(''p'') ≥ ''α'' for all ord ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Group Of Finite Morley Rank
In model theory, a stable group is a group that is stable in the sense of stability theory. An important class of examples is provided by groups of finite Morley rank (see below). Examples *A group of finite Morley rank is an abstract group ''G'' such that the formula ''x'' = ''x'' has finite Morley rank for the model ''G''. It follows from the definition that the theory of a group of finite Morley rank is ω-stable; therefore groups of finite Morley rank are stable groups. Groups of finite Morley rank behave in certain ways like finite-dimensional objects. The striking similarities between groups of finite Morley rank and finite groups are an object of active research. *All finite groups have finite Morley rank, in fact rank 0. *Algebraic groups over algebraically closed fields have finite Morley rank, equal to their dimension as algebraic sets. * showed that free groups, and more generally torsion-free hyperbolic groups, are stable. Free groups on more than one genera ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cherlin–Zilber Conjecture
In model theory, a stable group is a group that is stable in the sense of stability theory. An important class of examples is provided by groups of finite Morley rank (see below). Examples *A group of finite Morley rank is an abstract group ''G'' such that the formula ''x'' = ''x'' has finite Morley rank for the model ''G''. It follows from the definition that the theory of a group of finite Morley rank is ω-stable; therefore groups of finite Morley rank are stable groups. Groups of finite Morley rank behave in certain ways like finite-dimensional objects. The striking similarities between groups of finite Morley rank and finite groups are an object of active research. *All finite groups have finite Morley rank, in fact rank 0. *Algebraic groups over algebraically closed fields have finite Morley rank, equal to their dimension as algebraic sets. * showed that free groups, and more generally torsion-free hyperbolic groups, are stable. Free groups on more than one genera ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Disjoint Union
In mathematics, a disjoint union (or discriminated union) of a family of sets (A_i : i\in I) is a set A, often denoted by \bigsqcup_ A_i, with an injection of each A_i into A, such that the images of these injections form a partition of A (that is, each element of A belongs to exactly one of these images). A disjoint union of a family of pairwise disjoint sets is their union. In category theory, the disjoint union is the coproduct of the category of sets, and thus defined up to a bijection. In this context, the notation \coprod_ A_i is often used. The disjoint union of two sets A and B is written with infix notation as A \sqcup B. Some authors use the alternative notation A \uplus B or A \operatorname B (along with the corresponding \biguplus_ A_i or \operatorname_ A_i). A standard way for building the disjoint union is to define A as the set of ordered pairs (x, i) such that x \in A_i, and the injection A_i \to A as x \mapsto (x, i). Example Consider the sets A_0 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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. 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 comparison. Two elements ''x'' and ''y'' may stand in any of four mutually exclusive relationships to each other: either ''x'' ''y'', or ''x'' and ''y'' are ''incompar ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rational Numbers
In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all rational numbers, also referred to as "the rationals", the field of rationals or the field of rational numbers is usually denoted by boldface , or blackboard bold \mathbb. A rational number is a real number. The real numbers that are rational are those whose decimal expansion either terminates after a finite number of digits (example: ), or eventually begins to repeat the same finite sequence of digits over and over (example: ). This statement is true not only in base 10, but also in every other integer base, such as the binary and hexadecimal ones (see ). A real number that is not rational is called irrational. Irrational numbers include , , , and . Since the set of rational numbers is countable, and the set of real numbers is uncountable, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Degree (algebraic Geometry)
In mathematics, the degree of an affine or projective variety of dimension is the number of intersection points of the variety with hyperplanes in general position.In the affine case, the general-position hypothesis implies that there is no intersection point at infinity. For an algebraic set, the intersection points must be counted with their intersection multiplicity, because of the possibility of multiple components. For (irreducible) varieties, if one takes into account the multiplicities and, in the affine case, the points at infinity, the hypothesis of ''general position'' may be replaced by the much weaker condition that the intersection of the variety has the dimension zero (that is, consists of a finite number of points). This is a generalization of Bézout's theorem (For a proof, see ). The degree is not an intrinsic property of the variety, as it depends on a specific embedding of the variety in an affine or projective space. The degree of a hypersurface is equal to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Irreducible Component
In algebraic geometry, an irreducible algebraic set or irreducible variety is an algebraic set that cannot be written as the union of two proper algebraic subsets. An irreducible component is an algebraic subset that is irreducible and maximal (for set inclusion) for this property. For example, the set of solutions of the equation is not irreducible, and its irreducible components are the two lines of equations and . It is a fundamental theorem of classical algebraic geometry that every algebraic set may be written in a unique way as a finite union of irreducible components. These concepts can be reformulated in purely topological terms, using the Zariski topology, for which the closed sets are the algebraic subsets: A topological space is ''irreducible'' if it is not the union of two proper closed subsets, and an ''irreducible component'' is a maximal subspace (necessarily closed) that is irreducible for the induced topology. Although these concepts may be considered for every t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
