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Jónsson–Tarski Algebra
In mathematics, a Jónsson–Tarski algebra or Cantor algebra is an algebraic structure encoding a bijection from an infinite set onto the Cartesian product, product . They were introduced by . , named them after Georg Cantor because of Cantor's pairing function and Cantor's theorem that an infinite set has the same number of elements as . The term ''Cantor algebra'' is also occasionally used to mean the Boolean algebra (structure), Boolean algebra of all clopen subsets of the Cantor set, or the Boolean algebra of Borel subsets of the real number, reals modulo meager sets (sometimes called the Cohen algebra). The group of Order isomorphism, order-preserving automorphisms of the free object, free Jónsson–Tarski algebra on one generator (mathematics), generator is the Thompson groups, Thompson group . Definition A Jónsson–Tarski algebra of type 2 is a set with a product from to and two 'projection' maps and from to , satisfying , , and . The definition for type >&nbs ... [...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] |
Borel Subset
In mathematics, a Borel set is any set in a topological space that can be formed from open sets (or, equivalently, from closed sets) through the operations of countable union, countable intersection, and relative complement. Borel sets are named after Émile Borel. For a topological space ''X'', the collection of all Borel sets on ''X'' forms a σ-algebra, known as the Borel algebra or Borel σ-algebra. The Borel algebra on ''X'' is the smallest σ-algebra containing all open sets (or, equivalently, all closed sets). Borel sets are important in measure theory, since any measure defined on the open sets of a space, or on the closed sets of a space, must also be defined on all Borel sets of that space. Any measure defined on the Borel sets is called a Borel measure. Borel sets and the associated Borel hierarchy also play a fundamental role in descriptive set theory. In some contexts, Borel sets are defined to be generated by the compact sets of the topological space, r ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Thompson Groups
In mathematics, the Thompson groups (also called Thompson's groups, vagabond groups or chameleon groups) are three groups, commonly denoted F \subseteq T \subseteq V, that were introduced by Richard Thompson in some unpublished handwritten notes in 1965 as a possible counterexample to the von Neumann conjecture. Of the three, ''F'' is the most widely studied, and is sometimes referred to as the Thompson group or Thompson's group. The Thompson groups, and ''F'' in particular, have a collection of unusual properties that have made them counterexamples to many general conjectures in group theory. All three Thompson groups are infinite but finitely presented. The groups ''T'' and ''V'' are (rare) examples of infinite but finitely-presented simple groups. The group ''F'' is not simple but its derived subgroup 'F'',''F''is and the quotient of ''F'' by its derived subgroup is the free abelian group of rank 2. ''F'' is totally ordered, has exponential growth, and does not contain a subg ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Generator (mathematics)
In mathematics and physics, the term generator or generating set may refer to any of a number of related concepts. The underlying concept in each case is that of a smaller set of objects, together with a set of operations that can be applied to it, that result in the creation of a larger collection of objects, called the generated set. The larger set is then said to be generated by the smaller set. It is commonly the case that the generating set has a simpler set of properties than the generated set, thus making it easier to discuss and examine. It is usually the case that properties of the generating set are in some way preserved by the act of generation; likewise, the properties of the generated set are often reflected in the generating set. List of generators A list of examples of generating sets follow. * Generating set or spanning set of a vector space: a set that spans the vector space * Generating set of a group: A subset of a group that is not contained in any subgro ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Free Object
In mathematics, the idea of a free object is one of the basic concepts of abstract algebra. Informally, a free object over a set ''A'' can be thought of as being a "generic" algebraic structure over ''A'': the only equations that hold between elements of the free object are those that follow from the defining axioms of the algebraic structure. Examples include free groups, tensor algebras, or free lattices. The concept is a part of universal algebra, in the sense that it relates to all types of algebraic structure (with finitary operations). It also has a formulation in terms of category theory, although this is in yet more abstract terms. Definition Free objects are the direct generalization to categories of the notion of basis in a vector space. A linear function between vector spaces is entirely determined by its values on a basis of the vector space The following definition translates this to any category. A concrete category is a category that is equipped with a faithf ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Order Isomorphism
In the mathematical field of order theory, an order isomorphism is a special kind of monotone function that constitutes a suitable notion of isomorphism for partially ordered sets (posets). Whenever two posets are order isomorphic, they can be considered to be "essentially the same" in the sense that either of the orders can be obtained from the other just by renaming of elements. Two strictly weaker notions that relate to order isomorphisms are order embeddings and Galois connections. Definition Formally, given two posets (S,\le_S) and (T,\le_T), an order isomorphism from (S,\le_S) to (T,\le_T) is a bijective function f from S to T with the property that, for every x and y in S, x \le_S y if and only if f(x)\le_T f(y). That is, it is a bijective order-embedding. It is also possible to define an order isomorphism to be a surjective order-embedding. The two assumptions that f cover all the elements of T and that it preserve orderings, are enough to ensure that f is also one-to-one ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cohen Algebra
In mathematical set theory, a Cohen algebra, named after Paul Cohen, is a type of Boolean algebra used in the theory of forcing. A Cohen algebra is a Boolean algebra whose completion is isomorphic to the completion of a free Boolean algebra In mathematics, a free Boolean algebra is a Boolean algebra with a distinguished set of elements, called ''generators'', such that: #Each element of the Boolean algebra can be expressed as a finite combination of generators, using the Boolean opera ... . References * Forcing (mathematics) Boolean algebra {{Settheory-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Meager Set
In the mathematical field of general topology, a meagre set (also called a meager set or a set of first category) is a subset of a topological space that is small or negligible in a precise sense detailed below. A set that is not meagre is called nonmeagre, or of the second category. See below for definitions of other related terms. The meagre subsets of a fixed space form a σ-ideal of subsets; that is, any subset of a meagre set is meagre, and the union of countably many meagre sets is meagre. Meagre sets play an important role in the formulation of the notion of Baire space and of the Baire category theorem, which is used in the proof of several fundamental results of functional analysis. Definitions Throughout, X will be a topological space. A subset of X is called X, a of X, or of the in X if it is a countable union of nowhere dense subsets of X (where a nowhere dense set is a set whose closure has empty interior). The qualifier "in X" can be omitted if the ambient ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Real Number
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real number can be almost uniquely represented by an infinite decimal expansion. The real numbers are fundamental in calculus (and more generally in all mathematics), in particular by their role in the classical definitions of limits, continuity and derivatives. The set of real numbers is denoted or \mathbb and is sometimes called "the reals". The adjective ''real'' in this context was introduced in the 17th century by René Descartes to distinguish real numbers, associated with physical reality, from imaginary numbers (such as the square roots of ), which seemed like a theoretical contrivance unrelated to physical reality. The real numbers include the rational numbers, such as the integer and the fraction . The rest of the real number ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cantor Set
In mathematics, the Cantor set is a set of points lying on a single line segment that has a number of unintuitive properties. It was discovered in 1874 by Henry John Stephen Smith and introduced by German mathematician Georg Cantor in 1883. Through consideration of this set, Cantor and others helped lay the foundations of modern point-set topology. The most common construction is the Cantor ternary set, built by removing the middle third of a line segment and then repeating the process with the remaining shorter segments. Cantor mentioned the ternary construction only in passing, as an example of a more general idea, that of a perfect set that is nowhere dense. More generally, in topology, ''a'' Cantor space is a topological space homeomorphic to the Cantor ternary set (equipped with its subspace topology). By a theorem of Brouwer, this is equivalent to being perfect nonempty, compact metrizable and zero dimensional. Construction and formula of the ternary set The Cantor tern ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebraic Structure
In mathematics, an algebraic structure consists of a nonempty set ''A'' (called the underlying set, carrier set or domain), a collection of operations on ''A'' (typically binary operations such as addition and multiplication), and a finite set of identities, known as axioms, that these operations must satisfy. An algebraic structure may be based on other algebraic structures with operations and axioms involving several structures. For instance, a vector space involves a second structure called a field, and an operation called ''scalar multiplication'' between elements of the field (called '' scalars''), and elements of the vector space (called '' vectors''). Abstract algebra is the name that is commonly given to the study of algebraic structures. The general theory of algebraic structures has been formalized in universal algebra. Category theory is another formalization that includes also other mathematical structures and functions between structures of the same type (homomor ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Clopen Subset
In topology, a clopen set (a portmanteau of closed-open set) in a topological space is a set which is both open and closed. That this is possible may seem counter-intuitive, as the common meanings of and are antonyms, but their mathematical definitions are not mutually exclusive. A set is closed if its complement is open, which leaves the possibility of an open set whose complement is also open, making both sets both open closed, and therefore clopen. As described by topologist James Munkres, unlike a door, "a set can be open, or closed, or both, or neither!" emphasizing that the meaning of "open"/"closed" for is unrelated to their meaning for (and so the open/closed door dichotomy does not transfer to open/closed sets). This contrast to doors gave the class of topological spaces known as " door spaces" their name. Examples In any topological space X, the empty set and the whole space X are both clopen. Now consider the space X which consists of the union of the two op ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
